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Ingy döt Net 2023-07-01 11:58:00 -04:00
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3
Task/Bernoulli-numbers/00-META.yaml Normal file
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---
from: http://rosettacode.org/wiki/Bernoulli_numbers
note: Mathematics

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[[wp:Bernoulli number|Bernoulli numbers]] are used in some series expansions of several functions   (trigonometric, hyperbolic, gamma, etc.),   and are extremely important in number theory and analysis.
Note that there are two definitions of Bernoulli numbers;   this task will be using the modern usage   (as per   ''The National Institute of Standards and Technology convention'').
The &nbsp; n<sup>th</sup> &nbsp; Bernoulli number is expressed as &nbsp; '''B'''<sub>n</sub>.
<br>
;Task
:* &nbsp; show the Bernoulli numbers &nbsp; '''B'''<sub>0</sub> &nbsp; through &nbsp; '''B'''<sub>60</sub>.
:* &nbsp; suppress the output of values which are equal to zero. &nbsp; (Other than &nbsp; '''B'''<sub>1</sub>&nbsp;, all &nbsp; ''odd'' &nbsp; Bernoulli numbers have a value of zero.)
:* &nbsp; express the Bernoulli numbers as fractions &nbsp;(most are improper fractions).
:* &nbsp; the fractions should be reduced.
:* &nbsp; index each number in some way so that it can be discerned which Bernoulli number is being displayed.
:* &nbsp; align the solidi &nbsp; (<big><b>/</b></big>) &nbsp; if used &nbsp;(extra credit).
;An algorithm
The Akiyama–Tanigawa algorithm for the "second Bernoulli numbers" as taken from [[wp:Bernoulli_number#Algorithmic_description|wikipedia]] is as follows:
'''for''' ''m'' '''from''' 0 '''by''' 1 '''to''' ''n'' '''do'''
''A''[''m''] ← 1/(''m''+1)
'''for''' ''j'' '''from''' ''m'' '''by''' -1 '''to''' 1 '''do'''
''A''[''j''-1] ← ''j''×(''A''[''j''-1] - ''A''[''j''])
'''return''' ''A''[0] (which is ''B''<sub>''n''</sub>)
;See also
* Sequence [[oeis:A027641|A027641 Numerator of Bernoulli number B_n]] on The On-Line Encyclopedia of Integer Sequences.
* Sequence [[oeis:A027642|A027642 Denominator of Bernoulli number B_n]] on The On-Line Encyclopedia of Integer Sequences.
* Entry [http://mathworld.wolfram.com/BernoulliNumber.html Bernoulli number] on The Eric Weisstein's World of Mathematics (TM).
* Luschny's [http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html The Bernoulli Manifesto] for a discussion on &nbsp; <big> '''B<sub>1</sub> &nbsp; = &nbsp; -&frac12;''' &nbsp; versus &nbsp; '''+&frac12;'''. </big>
<br><br>

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Task/Bernoulli-numbers/ALGOL-68/bernoulli-numbers.alg Normal file
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BEGIN
# Show Bernoulli numbers B0 to B60 as rational numbers #
# Uses code from the Arithmetic/Rational task modified to use #
# LONG LONG INT to allow for the large number of digits requried #
PR precision 100 PR # sets the precision of LONG LONG INT #
# Code from the Arithmetic/Rational task #
# ============================================================== #
MODE FRAC = STRUCT( LONG LONG INT num #erator#, den #ominator#);
PROC gcd = (LONG LONG INT a, b) LONG LONG INT: # greatest common divisor #
(a = 0 | b |: b = 0 | a |: ABS a > ABS b | gcd(b, a MOD b) | gcd(a, b MOD a));
PROC lcm = (LONG LONG INT a, b)LONG LONG INT: # least common multiple #
a OVER gcd(a, b) * b;
PRIO // = 9; # higher then the ** operator #
OP // = (LONG LONG INT num, den)FRAC: ( # initialise and normalise #
LONG LONG INT common = gcd(num, den);
IF den < 0 THEN
( -num OVER common, -den OVER common)
ELSE
( num OVER common, den OVER common)
FI
);
OP + = (FRAC a, b)FRAC: (
LONG LONG INT common = lcm(den OF a, den OF b);
FRAC result := ( common OVER den OF a * num OF a + common OVER den OF b * num OF b, common );
num OF result//den OF result
);
OP - = (FRAC a, b)FRAC: a + -b,
* = (FRAC a, b)FRAC: (
LONG LONG INT num = num OF a * num OF b,
den = den OF a * den OF b;
LONG LONG INT common = gcd(num, den);
(num OVER common) // (den OVER common)
);
OP - = (FRAC frac)FRAC: (-num OF frac, den OF frac);
# ============================================================== #
# end code from the Arithmetic/Rational task #
# Additional FRACrelated operators #
OP * = ( INT a, FRAC b )FRAC: ( num OF b * a ) // den OF b;
OP // = ( INT a, INT b )FRAC: LONG LONG INT( a ) // LONG LONG INT( b );
# returns the nth Bernoulli number, n must be >= 0 #
# Uses the algorithm suggested by the task, so B(1) is +1/2 #
PROC bernoulli = ( INT n )FRAC:
IF n < 0
THEN # n is out of range # 0 // 1
ELSE # n is valid #
[ 0 : n ]FRAC a;
FOR i FROM LWB a TO UPB a DO a[ i ] := 0 // 1 OD;
FOR m FROM 0 TO n DO
a[ m ] := 1 // ( m + 1 );
FOR j FROM m BY -1 TO 1 DO
a[ j - 1 ] := j * ( a[ j - 1 ] - a[ j ] )
OD
OD;
a[ 0 ]
FI # bernoulli # ;
FOR n FROM 0 TO 60 DO
FRAC bn := bernoulli( n );
IF num OF bn /= 0 THEN
# have a non-0 Bn #
print( ( "B(", whole( n, -2 ), ") ", whole( num OF bn, -50 ), " / ", whole( den OF bn, 0 ), newline ) )
FI
OD
END

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Task/Bernoulli-numbers/Ada/bernoulli-numbers.ada Normal file
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WITH GMP.Rationals, GMP.Integers, Ada.Text_IO, Ada.Strings.Fixed, Ada.Strings;
USE GMP.Rationals, GMP.Integers, Ada.Text_IO, Ada.Strings.Fixed, Ada.Strings;
PROCEDURE Main IS
FUNCTION Bernoulli_Number (N : Natural) RETURN Unbounded_Fraction IS
FUNCTION "/" (Left, Right : Natural) RETURN Unbounded_Fraction IS
(To_Unbounded_Integer (Left) / To_Unbounded_Integer (Right));
A : ARRAY (0 .. N) OF Unbounded_Fraction;
BEGIN
FOR M IN 0 .. N LOOP
A (M) := 1 / (M + 1);
FOR J IN REVERSE 1 .. M LOOP
A (J - 1) := (J / 1 ) * (A (J - 1) - A (J));
END LOOP;
END LOOP;
RETURN A (0);
END Bernoulli_Number;
BEGIN
FOR I IN 0 .. 60 LOOP
IF I MOD 2 = 0 OR I = 1 THEN
DECLARE
B : Unbounded_Fraction := Bernoulli_Number (I);
S : String := Image (GMP.Rationals.Numerator (B));
BEGIN
Put_Line ("B (" & (IF I < 10 THEN " " ELSE "") & Trim (I'Img, Left)
& ")=" & (44 - S'Length) * " " & Image (B));
END;
END IF;
END LOOP;
END Main;

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Task/Bernoulli-numbers/AppleScript/bernoulli-numbers-1.applescript Normal file
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on bernoullis(n) -- Return a list of "numerator / denominator" texts representing Bernoulli numbers B(0) to B(n).
set listMathScript to getListMathScript(10) -- Script object providing custom list math routines.
set output to {}
-- Akiyama–Tanigawa algorithm for the "second Bernoulli numbers".
-- List 'a' will contain {numerator, denominator} lists representing fractions.
-- The numerators and denominators will in turn be lists containing integers representing their (decimal) digits.
set a to {}
repeat with m from 0 to n
-- Append the structure for 1 / (m + 1) to the end of a.
set {numerator2, denominator2} to {{1}, listMathScript's intToList(m + 1)}
set a's end to result
repeat with j from m to 1 by -1
-- Retrieve the preceding numerator and denominator.
set {numerator1, denominator1} to a's item j
tell listMathScript
-- Get the two fractions' lowest common denominator and adjust the numerators accordingly.
set lcd to its lcm(denominator1, denominator2)
set numerator1 to its multiply(numerator1, its |div|(lcd, denominator1))
set numerator2 to its multiply(numerator2, its |div|(lcd, denominator2))
-- Subtract numerator2 from numerator1 and multiply the result by j.
-- Assign the results to numerator2 and denominator2 for the next iteration.
set numerator2 to its multiply(its subtract(numerator1, numerator2), its intToList(j))
set denominator2 to lcd
end tell
-- Also store them in a's slot j. No need to reduce them here.
set a's item j to {numerator2, denominator2}
end repeat
-- The fraction just stored in a's first slot is Bernoulli(m). Reduce it and append a text representation to the output.
tell listMathScript
set gcd to its hcf(numerator2, denominator2)
set numerator2 to its |div|(numerator2, gcd)
set denominator2 to its |div|(denominator2, gcd)
set end of output to its listToText(numerator2) & (" / " & its listToText(denominator2))
end tell
end repeat
return output
end bernoullis
on getListMathScript(base)
script
on multiply(lst1, lst2) -- Multiply lst1 by lst2.
set lst1Length to (count lst1)
set lst2Length to (count lst2)
set productLength to lst1Length + lst2Length - 1
set product to {}
repeat productLength times
set product's end to 0
end repeat
-- Long multiplication algorithm, updating product digits on the fly instead of summing rows at the end.
repeat with lst2Index from -1 to -lst2Length by -1
set lst2Digit to lst2's item lst2Index
if (lst2Digit is not 0) then
set carry to 0
set productIndex to lst2Index
repeat with lst1Index from lst1's length to 1 by -1
tell lst2Digit * (lst1's item lst1Index) + carry + (product's item productIndex)
set product's item productIndex to (it mod base)
set carry to (it div base)
end tell
set productIndex to productIndex - 1
end repeat
if (carry = 0) then
else if (productIndex < -productLength) then
set product's beginning to carry
else
set product's item productIndex to (product's item productIndex) + carry
end if
end if
end repeat
return product
end multiply
on subtract(lst1, lst2) -- Subtract lst2 from lst1.
set lst1Length to (count lst1)
set lst2Length to (count lst2)
-- Pad copies to equal lengths.
copy lst1 to lst1
repeat (lst2Length - lst1Length) times
set lst1's beginning to 0
end repeat
copy lst2 to lst2
repeat (lst1Length - lst2Length) times
set lst2's beginning to 0
end repeat
-- Is lst2's numeric value greater than lst1's?
set paddedLength to (count lst1)
repeat with i from 1 to paddedLength
set lst1Digit to lst1's item i
set lst2Digit to lst2's item i
set lst2Greater to (lst2Digit > lst1Digit)
if ((lst2Greater) or (lst1Digit > lst2Digit)) then exit repeat
end repeat
-- If so, set up to subtract lst1 from lst2 instead. We'll invert the result's sign at the end.
if (lst2Greater) then tell lst2
set lst2 to lst1
set lst1 to it
end tell
-- The subtraction at last!
set difference to {}
set borrow to 0
repeat with i from paddedLength to 1 by -1
tell (lst1's item i) + base - borrow - (lst2's item i)
set difference's beginning to (it mod base)
set borrow to 1 - (it div base)
end tell
end repeat
if (lst2Greater) then invert(difference)
return difference
end subtract
on |div|(lst1, lst2) -- List lst1 div lst2.
return divide(lst1, lst2)'s quotient
end |div|
on |mod|(lst1, lst2) -- List lst1 mod lst2.
return divide(lst1, lst2)'s remainder
end |mod|
on divide(lst1, lst2) -- Divide lst1 by lst2. Return a record containing separate lists for the quotient and remainder.
set dividend to trim(lst1)
set divisor to trim(lst2)
set dividendLength to (count dividend)
set divisorLength to (count divisor)
if (divisorLength > dividendLength) then return {quotient:{0}, remainder:dividend}
-- Note the dividend's and divisor's signs, but use absolute values in the division.
set dividendNegative to (dividend's beginning < 0)
if (dividendNegative) then invert(dividend)
set divisorNegative to (divisor's beginning < 0)
if (divisorNegative) then invert(divisor)
-- Long-division algorithm, but quotient digits are subtraction counts.
set quotient to {}
if (divisorLength > 1) then
set remainder to dividend's items 1 thru (divisorLength - 1)
else
set remainder to {}
end if
repeat with nextSlot from divisorLength to dividendLength
set remainder's end to dividend's item nextSlot
repeat with subtractionCount from 0 to base -- Only ever reaches base - 1.
set subtractionResult to trim(subtract(remainder, divisor))
if (subtractionResult's beginning < 0) then exit repeat
set remainder to subtractionResult
end repeat
set end of quotient to subtractionCount
end repeat
-- The quotient's negative if the input signs are different. Positive otherwise.
if (dividendNegative ≠ divisorNegative) then invert(quotient)
-- The remainder has the same sign as the dividend.
if (dividendNegative) then invert(remainder)
return {quotient:quotient, remainder:remainder}
end divide
on lcm(lst1, lst2) -- Lowest common multiple of lst1 and lst2.
return multiply(lst2, |div|(lst1, hcf(lst1, lst2)))
end lcm
on hcf(lst1, lst2) -- Highest common factor of lst1 and lst2.
set lst1 to trim(lst1)
set lst2 to trim(lst2)
repeat until (lst2 = {0})
set x to lst1
set lst1 to lst2
set lst2 to trim(|mod|(x, lst2))
end repeat
if (lst1's beginning < 0) then invert(lst1)
return lst1
end hcf
on invert(lst) -- Invert the sign of all lst's "digits".
repeat with thisDigit in lst
set thisDigit's contents to -thisDigit
end repeat
end invert
on trim(lst) -- Return a copy of lst with no leading zeros.
repeat with i from 1 to (count lst)
if (lst's item i is not 0) then exit repeat
end repeat
return lst's items i thru end
end trim
on intToList(n) -- Return a list of numbers representing n's digits.
set lst to {n mod base}
set n to n div base
repeat until (n = 0)
set beginning of lst to n mod base as integer
set n to n div base
end repeat
return lst
end intToList
on listToText(lst) -- Return the number represented by the input list as text.
-- This lazily assumes 2 <= base <= 10. :)
set lst to trim(lst)
if (lst's beginning < 0) then
invert(lst)
set lst's beginning to "-"
end if
return join(lst, "")
end listToText
end script
return result
end getListMathScript
on join(lst, delim)
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to delim
set txt to lst as text
set AppleScript's text item delimiters to astid
return txt
end join
on task()
set maxN to 60
set output to {""}
set padding to " = "
set bernoulliNumbers to bernoullis(maxN)
repeat with n from 0 to maxN
set bernie to bernoulliNumbers's item (n + 1)
if (bernie does not start with "0") then
set Bn to "B(" & n & ")"
set output's end to Bn & ¬
text ((count Bn) - 3) thru (50 - (offset of "/" in bernie)) of padding & ¬
bernie
end if
end repeat
return join(output, linefeed)
end task
task()

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Task/Bernoulli-numbers/AppleScript/bernoulli-numbers-2.applescript Normal file
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"
B(0) = 1 / 1
B(1) = 1 / 2
B(2) = 1 / 6
B(4) = -1 / 30
B(6) = 1 / 42
B(8) = -1 / 30
B(10) = 5 / 66
B(12) = -691 / 2730
B(14) = 7 / 6
B(16) = -3617 / 510
B(18) = 43867 / 798
B(20) = -174611 / 330
B(22) = 854513 / 138
B(24) = -236364091 / 2730
B(26) = 8553103 / 6
B(28) = -23749461029 / 870
B(30) = 8615841276005 / 14322
B(32) = -7709321041217 / 510
B(34) = 2577687858367 / 6
B(36) = -26315271553053477373 / 1919190
B(38) = 2929993913841559 / 6
B(40) = -261082718496449122051 / 13530
B(42) = 1520097643918070802691 / 1806
B(44) = -27833269579301024235023 / 690
B(46) = 596451111593912163277961 / 282
B(48) = -5609403368997817686249127547 / 46410
B(50) = 495057205241079648212477525 / 66
B(52) = -801165718135489957347924991853 / 1590
B(54) = 29149963634884862421418123812691 / 798
B(56) = -2479392929313226753685415739663229 / 870
B(58) = 84483613348880041862046775994036021 / 354
B(60) = -1215233140483755572040304994079820246041491 / 56786730"

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Task/Bernoulli-numbers/Bracmat/bernoulli-numbers.bracmat Normal file
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( BernoulliList
= B Bs answer indLn indexLen indexPadding
, n numberPadding p solPos solidusPos sp
. ( B
= m A a j b
. -1:?m
& :?A
& whl
' ( 1+!m:~>!arg:?m
& ((!m+1:?j)^-1:?a)
map
$ ( (
= .(-1+!j:?j)*(!arg+-1*!a):?a
)
. !A
)
: ?A
)
& !A:? @?b
& !b
)
& -1:?n
& :?Bs
& whl
' ( 1+!n:~>!arg:?n
& B$!n !Bs:?Bs
)
& @(!arg:? [?indexLen)
& 1+!indexLen:?indexLen
& !Bs:%@(?:? "/" [?solidusPos ?) ?
& 1+!solidusPos:?solidusPos:?p
& :?sp
& whl
' (!p+-1:~<0:?p&" " !sp:?sp)
& :?answer
& whl
' ( !Bs:%?B ?Bs
& ( !B:0
| (!B:/|str$(!B "/1"):?B)
& @(!B:? "/" [?solPos ?)
& @(!arg:? [?indLn)
& !sp
: ? [(-1*!indexLen+!indLn) ?indexPadding
: ? [(-1*!solidusPos+!solPos) ?numberPadding
& "B("
!arg
")="
!indexPadding
!numberPadding
(!B:>0&" "|)
!B
\n
!answer
: ?answer
)
& -1+!arg:?arg
)
& str$!answer
)
& BernoulliList$60;

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Task/Bernoulli-numbers/C++/bernoulli-numbers.cpp Normal file
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/**
* Configured with: --prefix=/Library/Developer/CommandLineTools/usr --with-gxx-include-dir=/usr/include/c++/4.2.1
* Apple LLVM version 9.1.0 (clang-902.0.39.1)
* Target: x86_64-apple-darwin17.5.0
* Thread model: posix
*/
#include <boost/multiprecision/cpp_int.hpp> // 1024bit precision
#include <boost/rational.hpp> // Rationals
#include <iostream> // formatting with std::cout
#include <vector> // Container
typedef boost::rational<boost::multiprecision::int1024_t> rational; // reduce boilerplate
rational bernoulli(size_t n) {
auto out = std::vector<rational>();
for (size_t m = 0; m <= n; m++) {
out.emplace_back(1, (m + 1)); // automatically constructs object
for (size_t j = m; j >= 1; j--) {
out[j - 1] = rational(j) * (out[j - 1] - out[j]);
}
}
return out[0];
}
int main() {
for (size_t n = 0; n <= 60; n += n >= 2 ? 2 : 1) {
auto b = bernoulli(n);
std::cout << "B(" << std::right << std::setw(2) << n << ") = ";
std::cout << std::right << std::setw(44) << b.numerator();
std::cout << " / " << b.denominator() << std::endl;
}
return 0;
}

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Task/Bernoulli-numbers/C-sharp/bernoulli-numbers-1.cs Normal file
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using Mpir.NET;
using System;
namespace Bernoulli
{
class Program
{
private static void bernoulli(mpq_t rop, uint n)
{
mpq_t[] a = new mpq_t[n + 1];
for (uint i = 0; i < n + 1; i++)
{
a[i] = new mpq_t();
}
for (uint m = 0; m <= n; ++m)
{
mpir.mpq_set_ui(a[m], 1, m + 1);
for (uint j = m; j > 0; --j)
{
mpir.mpq_sub(a[j - 1], a[j], a[j - 1]);
mpir.mpq_set_ui(rop, j, 1);
mpir.mpq_mul(a[j - 1], a[j - 1], rop);
}
mpir.mpq_set(rop, a[0]);
}
}
static void Main(string[] args)
{
mpq_t rop = new mpq_t();
mpz_t n = new mpz_t();
mpz_t d = new mpz_t();
for (uint i = 0; i <= 60; ++i)
{
bernoulli(rop, i);
if (mpir.mpq_cmp_ui(rop, 0, 1) != 0)
{
mpir.mpq_get_num(n, rop);
mpir.mpq_get_den(d, rop);
Console.WriteLine(string.Format("B({0, 2}) = {1, 44} / {2}", i, n, d));
}
}
Console.ReadKey();
}
}
}

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Task/Bernoulli-numbers/C-sharp/bernoulli-numbers-2.cs Normal file
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using System;
using System.Console;
using System.Linq;
using MathNet.Numerics;
namespace Rosettacode.Rational.CS
{
class Program
{
private static readonly Func<int, BigRational> ℚ = BigRational.FromInt;
private static BigRational CalculateBernoulli(int n)
{
var a = InitializeArray(n);
foreach(var m in Enumerable.Range(1,n))
{
a[m] = ℚ(1) / (ℚ(m) + ℚ(1));
for (var j = m; j >= 1; j--)
{
a[j-1] = ℚ(j) * (a[j-1] - a[j]);
}
}
return a[0];
}
private static BigRational[] InitializeArray(int n)
{
var a = new BigRational[n + 1];
for (var x = 0; x < a.Length; x++)
{
a[x] = ℚ(x + 1);
}
return a;
}
static void Main()
{
Enumerable.Range(0, 61) // the second parameter is the number of range elements, and is not the final item of the range.
.Select(n => new {N = n, BernoulliNumber = CalculateBernoulli(n)})
.Where(b => !b.BernoulliNumber.Numerator.IsZero)
.Select(b => string.Format("B({0, 2}) = {1, 44} / {2}", b.N, b.BernoulliNumber.Numerator, b.BernoulliNumber.Denominator))
.ToList()
.ForEach(WriteLine);
}
}
}

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Task/Bernoulli-numbers/C-sharp/bernoulli-numbers-3.cs Normal file
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using System;
using System.Numerics;
using System.Collections.Generic;
namespace bern
{
class Program
{
struct BerNum { public int index; public BigInteger Numer, Denomin; };
static int w1 = 1, w2 = 1; // widths for formatting output
static int max = 60; // default maximum, can override on command line
// returns nth Bernoulli number
static BerNum CalcBernoulli(int n)
{
BerNum res;
BigInteger f;
BigInteger[] nu = new BigInteger[n + 1],
de = new BigInteger[n + 1];
for (int m = 0; m <= n; m++)
{
nu[m] = 1; de[m] = m + 1;
for (int j = m; j > 0; j--)
if ((f = BigInteger.GreatestCommonDivisor(
nu[j - 1] = j * (de[j] * nu[j - 1] - de[j - 1] * nu[j]),
de[j - 1] *= de[j])) != BigInteger.One)
{ nu[j - 1] /= f; de[j - 1] /= f; }
}
res.index = n; res.Numer = nu[0]; res.Denomin = de[0];
w1 = Math.Max(n.ToString().Length, w1); // ratchet up widths
w2 = Math.Max(res.Numer.ToString().Length, w2);
if (max > 50) Console.Write("."); // progress dots appear for larger values
return res;
}
static void Main(string[] args)
{
List<BerNum> BNumbList = new List<BerNum>();
// defaults to 60 when no (or invalid) command line parameter is present
if (args.Length > 0) {
int.TryParse(args[0], out max);
if (max < 1 || max > Int16.MaxValue) max = 60;
if (args[0] == "0") max = 0;
}
for (int i = 0; i <= max; i++) // fill list with values
{
BerNum BNumb = CalcBernoulli(i);
if (BNumb.Numer != BigInteger.Zero) BNumbList.Add(BNumb);
}
if (max > 50) Console.WriteLine();
string strFmt = "B({0, " + w1.ToString() + "}) = {1, " + w2.ToString() + "} / {2}";
// display formatted list
foreach (BerNum bn in BNumbList)
Console.WriteLine(strFmt , bn.index, bn.Numer, bn.Denomin);
if (System.Diagnostics.Debugger.IsAttached) Console.Read();
}
}
}

51
Task/Bernoulli-numbers/C/bernoulli-numbers.c Normal file
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#include <stdlib.h>
#include <gmp.h>
#define mpq_for(buf, op, n)\
do {\
size_t i;\
for (i = 0; i < (n); ++i)\
mpq_##op(buf[i]);\
} while (0)
void bernoulli(mpq_t rop, unsigned int n)
{
unsigned int m, j;
mpq_t *a = malloc(sizeof(mpq_t) * (n + 1));
mpq_for(a, init, n + 1);
for (m = 0; m <= n; ++m) {
mpq_set_ui(a[m], 1, m + 1);
for (j = m; j > 0; --j) {
mpq_sub(a[j-1], a[j], a[j-1]);
mpq_set_ui(rop, j, 1);
mpq_mul(a[j-1], a[j-1], rop);
}
}
mpq_set(rop, a[0]);
mpq_for(a, clear, n + 1);
free(a);
}
int main(void)
{
mpq_t rop;
mpz_t n, d;
mpq_init(rop);
mpz_inits(n, d, NULL);
unsigned int i;
for (i = 0; i <= 60; ++i) {
bernoulli(rop, i);
if (mpq_cmp_ui(rop, 0, 1)) {
mpq_get_num(n, rop);
mpq_get_den(d, rop);
gmp_printf("B(%-2u) = %44Zd / %Zd\n", i, n, d);
}
}
mpz_clears(n, d, NULL);
mpq_clear(rop);
return 0;
}

24
Task/Bernoulli-numbers/Clojure/bernoulli-numbers.clj Normal file
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ns test-project-intellij.core
(:gen-class))
(defn a-t [n]
" Used Akiyama-Tanigawa algorithm with a single loop rather than double nested loop "
" Clojure does fractional arithmetic automatically so that part is easy "
(loop [m 0
j m
A (vec (map #(/ 1 %) (range 1 (+ n 2))))] ; Prefil A(m) with 1/(m+1), for m = 1 to n
(cond ; Three way conditional allows single loop
(>= j 1) (recur m (dec j) (assoc A (dec j) (* j (- (nth A (dec j)) (nth A j))))) ; A[j-1] ← j×(A[j-1] - A[j]) ;
(< m n) (recur (inc m) (inc m) A) ; increment m, reset j = m
:else (nth A 0))))
(defn format-ans [ans]
" Formats answer so that '/' is aligned for all answers "
(if (= ans 1)
(format "%50d / %8d" 1 1)
(format "%50d / %8d" (numerator ans) (denominator ans))))
;; Generate a set of results for [0 1 2 4 ... 60]
(doseq [q (flatten [0 1 (range 2 62 2)])
:let [ans (a-t q)]]
(println q ":" (format-ans ans)))

43
Task/Bernoulli-numbers/Common-Lisp/bernoulli-numbers.lisp Normal file
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(defun bernouilli (n)
(loop with a = (make-array (list (1+ n)))
for m from 0 to n do
(setf (aref a m) (/ 1 (+ m 1)))
(loop for j from m downto 1 do
(setf (aref a (- j 1))
(* j (- (aref a j) (aref a (- j 1))))))
finally (return (aref a 0))))
;;Print outputs to stdout:
(loop for n from 0 to 60 do
(let ((b (bernouilli n)))
(when (not (zerop b))
(format t "~a: ~a~%" n b))))
;;For the "extra credit" challenge, we need to align the slashes.
(let (results)
;;collect the results
(loop for n from 0 to 60 do
(let ((b (bernouilli n)))
(when (not (zerop b)) (push (cons b n) results))))
;;parse the numerators into strings; save the greatest length in max-length
(let ((max-length (apply #'max (mapcar (lambda (r)
(length (format nil "~a" (numerator r))))
(mapcar #'car results)))))
;;Print the numbers with using the fixed-width formatter: ~Nd, where N is
;;the number of leading spaces. We can't just pass in the width variable
;;but we can splice together a formatting string that includes it.
;;We also can't use the fixed-width formatter on a ratio, so we have to split
;;the ratio and splice it back together like idiots.
(loop for n in (mapcar #'cdr (reverse results))
for r in (mapcar #'car (reverse results)) do
(format t (concatenate 'string
"B(~2d): ~"
(format nil "~a" max-length)
"d/~a~%")
n
(numerator r)
(denominator r)))))

27
Task/Bernoulli-numbers/Crystal/bernoulli-numbers-1.crystal Normal file
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require "big"
class Bernoulli
include Iterator(Tuple(Int32, BigRational))
def initialize
@a = [] of BigRational
@m = 0
end
def next
@a << BigRational.new(1, @m+1)
@m.downto(1) { |j| @a[j-1] = j*(@a[j-1] - @a[j]) }
v = @m.odd? && @m != 1 ? BigRational.new(0, 1) : @a.first
return {@m, v}
ensure
@m += 1
end
end
b = Bernoulli.new
bn = b.first(61).to_a
max_width = bn.map { |_, v| v.numerator.to_s.size }.max
bn.reject { |i, v| v.zero? }.each do |i, v|
puts "B(%2i) = %*i/%i" % [i, max_width, v.numerator, v.denominator]
end

14
Task/Bernoulli-numbers/Crystal/bernoulli-numbers-2.crystal Normal file
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require "big"
def bernoulli(n)
ar = [] of BigRational
(0..n).each do |m|
ar << BigRational.new(1, m+1)
m.downto(1) { |j| ar[j-1] = j * (ar[j-1] - ar[j]) }
end
ar[0] # (which is Bn)
end
b_nums = (0..61).map { |i| bernoulli(i) }
width = b_nums.map{ |b| b.numerator.to_s.size }.max
b_nums.each_with_index { |b,i| puts "B(%2i) = %*i/%i" % [i, width, b.numerator, b.denominator] unless b.zero? }

15
Task/Bernoulli-numbers/Crystal/bernoulli-numbers-3.crystal Normal file
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require "big"
def bernoulli2(limit)
ar = [] of BigRational
(0..limit).each do |m|
ar << BigRational.new(1, m+1)
m.downto(1) { |j| ar[j-1] = j * (ar[j-1] - ar[j]) }
yield ar[0] # use Bn value in required block
end
end
b_nums = [] of BigRational
bernoulli2(61){ |b| b_nums << b }
width = b_nums.map{ |b| b.numerator.to_s.size }.max
b_nums.each_with_index { |b,i| puts "B(%2i) = %*i/%i" % [i, width, b.numerator, b.denominator] unless b.zero? }

18
Task/Bernoulli-numbers/D/bernoulli-numbers.d Normal file
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import std.stdio, std.range, std.algorithm, std.conv, arithmetic_rational;
auto bernoulli(in uint n) pure nothrow /*@safe*/ {
auto A = new Rational[n + 1];
foreach (immutable m; 0 .. n + 1) {
A[m] = Rational(1, m + 1);
foreach_reverse (immutable j; 1 .. m + 1)
A[j - 1] = j * (A[j - 1] - A[j]);
}
return A[0];
}
void main() {
immutable berns = 61.iota.map!bernoulli.enumerate.filter!(t => t[1]).array;
immutable width = berns.map!(b => b[1].numerator.text.length).reduce!max;
foreach (immutable b; berns)
writefln("B(%2d) = %*d/%d", b[0], width, b[1].tupleof);
}

32
Task/Bernoulli-numbers/Delphi/bernoulli-numbers.delphi Normal file
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program Bernoulli_numbers;
{$APPTYPE CONSOLE}
uses
System.SysUtils,
Velthuis.BigRationals;
function b(n: Integer): BigRational;
begin
var a: TArray<BigRational>;
SetLength(a, n + 1);
for var m := 0 to High(a) do
begin
a[m] := BigRational.Create(1, m + 1);
for var j := m downto 1 do
begin
a[j - 1] := (a[j - 1] - a[j]) * j;
end;
end;
Result := a[0];
end;
begin
for var n := 0 to 60 do
begin
var bb := b(n);
if bb.Numerator.BitLength > 0 then
writeln(format('B(%2d) =%45s/%s', [n, bb.Numerator.ToString, bb.Denominator.ToString]));
end;
readln;
end.

24
Task/Bernoulli-numbers/EchoLisp/bernoulli-numbers-1.l Normal file
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(lib 'bigint) ;; lerge numbers
(lib 'gloops) ;; classes
(define-class Rational null ((a :initform #0) (b :initform #1)))
(define-method tostring (Rational) (lambda (r) (format "%50d / %d" r.a r.b)))
(define-method normalize (Rational) (lambda (r) ;; divide a and b by gcd
(let ((g (gcd r.a r.b)))
(set! r.a (/ r.a g)) (set! r.b (/ r.b g))
(when (< r.b 0) (set! r.a ( - r.a)) (set! r.b (- r.b))) ;; denominator > 0
r)))
(define-method initialize (Rational) (lambda (r) (normalize r)))
(define-method add (Rational) (lambda (r n) ;; + Rational any number
(normalize (Rational (+ (* (+ #0 n) r.b) r.a) r.b))))
(define-method add (Rational Rational) (lambda (r q) ;;; + Rational Rational
(normalize (Rational (+ (* r.a q.b) (* r.b q.a)) (* r.b q.b)))))
(define-method sub (Rational Rational) (lambda (r q)
(normalize (Rational (- (* r.a q.b) (* r.b q.a)) (* r.b q.b)))))
(define-method mul (Rational Rational) (lambda (r q)
(normalize (Rational (* r.a q.a) (* r.b q.b)))))
(define-method mul (Rational) (lambda (r n)
(normalize (Rational (* r.a (+ #0 n)) r.b ))))
(define-method div (Rational Rational) (lambda (r q)
(normalize (Rational (* r.a q.b) (* r.b q.a)))))

47
Task/Bernoulli-numbers/EchoLisp/bernoulli-numbers-2.l Normal file
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;; Bernoulli numbers
;; http://rosettacode.org/wiki/Bernoulli_numbers
(define A (make-vector 100 0))
(define (B n)
(for ((m (1+ n))) ;; #1 creates a large integer
(vector-set! A m (Rational #1 (+ #1 m)))
(for ((j (in-range m 0 -1)))
(vector-set! A (1- j)
(mul (sub (vector-ref A (1- j)) (vector-ref A j)) j))))
(vector-ref A 0))
(for ((b (in-range 0 62 2))) (writeln b (B b))) →
0 1 / 1
2 1 / 6
4 -1 / 30
6 1 / 42
8 -1 / 30
10 5 / 66
12 -691 / 2730
14 7 / 6
16 -3617 / 510
18 43867 / 798
20 -174611 / 330
22 854513 / 138
24 -236364091 / 2730
26 8553103 / 6
28 -23749461029 / 870
30 8615841276005 / 14322
32 -7709321041217 / 510
34 2577687858367 / 6
36 -26315271553053477373 / 1919190
38 2929993913841559 / 6
40 -261082718496449122051 / 13530
42 1520097643918070802691 / 1806
44 -27833269579301024235023 / 690
46 596451111593912163277961 / 282
48 -5609403368997817686249127547 / 46410
50 495057205241079648212477525 / 66
52 -801165718135489957347924991853 / 1590
54 29149963634884862421418123812691 / 798
56 -2479392929313226753685415739663229 / 870
58 84483613348880041862046775994036021 / 354
60 -1215233140483755572040304994079820246041491 / 56786730
(B 1) → 1 / 2

55
Task/Bernoulli-numbers/Elixir/bernoulli-numbers.elixir Normal file
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defmodule Bernoulli do
defmodule Rational do
import Kernel, except: [div: 2]
defstruct numerator: 0, denominator: 1
def new(numerator, denominator\\1) do
sign = if numerator * denominator < 0, do: -1, else: 1
{numerator, denominator} = {abs(numerator), abs(denominator)}
gcd = gcd(numerator, denominator)
%Rational{numerator: sign * Kernel.div(numerator, gcd),
denominator: Kernel.div(denominator, gcd)}
end
def sub(a, b) do
new(a.numerator * b.denominator - b.numerator * a.denominator,
a.denominator * b.denominator)
end
def mul(a, b) when is_integer(a) do
new(a * b.numerator, b.denominator)
end
defp gcd(a,0), do: a
defp gcd(a,b), do: gcd(b, rem(a,b))
end
def numbers(n) do
Stream.transform(0..n, {}, fn m,acc ->
acc = Tuple.append(acc, Rational.new(1,m+1))
if m>0 do
new =
Enum.reduce(m..1, acc, fn j,ar ->
put_elem(ar, j-1, Rational.mul(j, Rational.sub(elem(ar,j-1), elem(ar,j))))
end)
{[elem(new,0)], new}
else
{[elem(acc,0)], acc}
end
end) |> Enum.to_list
end
def task(n \\ 61) do
b_nums = numbers(n)
width = Enum.map(b_nums, fn b -> b.numerator |> to_string |> String.length end)
|> Enum.max
format = 'B(~2w) = ~#{width}w / ~w~n'
Enum.with_index(b_nums)
|> Enum.each(fn {b,i} ->
if b.numerator != 0, do: :io.fwrite format, [i, b.numerator, b.denominator]
end)
end
end
Bernoulli.task

25
Task/Bernoulli-numbers/F-Sharp/bernoulli-numbers.fs Normal file
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open MathNet.Numerics
open System
open System.Collections.Generic
let calculateBernoulli n =
let ℚ(x) = BigRational.FromInt x
let A = Array.init<BigRational> (n+1) (fun x -> ℚ(x+1))
for m in [1..n] do
A.[m] <- ℚ(1) / (ℚ(m) + ℚ(1))
for j in [m..(-1)..1] do
A.[j-1] <- ℚ(j) * (A.[j-1] - A.[j])
A.[0]
[<EntryPoint>]
let main argv =
for n in [0..60] do
let bernoulliNumber = calculateBernoulli n
match bernoulliNumber.Numerator.IsZero with
| false ->
let formatedString = String.Format("B({0, 2}) = {1, 44} / {2}", n, bernoulliNumber.Numerator, bernoulliNumber.Denominator)
printfn "%s" formatedString
| true ->
printf ""
0

42
Task/Bernoulli-numbers/Factor/bernoulli-numbers-1.factor Normal file
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IN: scratchpad
[
0 1 1 "%2d : %d / %d\n" printf
1 -1 2 "%2d : %d / %d\n" printf
30 iota [
1 + 2 * dup bernoulli [ numerator ] [ denominator ] bi
"%2d : %d / %d\n" printf
] each
] time
0 : 1 / 1
1 : -1 / 2
2 : 1 / 6
4 : -1 / 30
6 : 1 / 42
8 : -1 / 30
10 : 5 / 66
12 : -691 / 2730
14 : 7 / 6
16 : -3617 / 510
18 : 43867 / 798
20 : -174611 / 330
22 : 854513 / 138
24 : -236364091 / 2730
26 : 8553103 / 6
28 : -23749461029 / 870
30 : 8615841276005 / 14322
32 : -7709321041217 / 510
34 : 2577687858367 / 6
36 : -26315271553053477373 / 1919190
38 : 2929993913841559 / 6
40 : -261082718496449122051 / 13530
42 : 1520097643918070802691 / 1806
44 : -27833269579301024235023 / 690
46 : 596451111593912163277961 / 282
48 : -5609403368997817686249127547 / 46410
50 : 495057205241079648212477525 / 66
52 : -801165718135489957347924991853 / 1590
54 : 29149963634884862421418123812691 / 798
56 : -2479392929313226753685415739663229 / 870
58 : 84483613348880041862046775994036021 / 354
60 : -1215233140483755572040304994079820246041491 / 56786730
Running time: 0.00489444 seconds

35
Task/Bernoulli-numbers/Factor/bernoulli-numbers-2.factor Normal file
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:: bernoulli-numbers ( n -- )
n 1 + 0 <array> :> tab
1 1 tab set-nth
2 n [a,b] [| k |
k 1 - dup
tab nth *
k tab set-nth
] each
2 n [a,b] [| k |
k n [a,b] [| j |
j tab nth
j k - 2 + *
j 1 - tab nth
j k - * +
j tab set-nth
] each
] each
1 :> s!
1 n [a,b] [| k |
k 2 * dup
2^ dup 1 - *
k tab nth
swap / *
s * k tab set-nth
s -1 * s!
] each
0 1 1 "%2d : %d / %d\n" printf
1 -1 2 "%2d : %d / %d\n" printf
1 n [a,b] [| k |
k 2 * k tab nth
[ numerator ] [ denominator ] bi
"%2d : %d / %d\n" printf
] each
;

3
Task/Bernoulli-numbers/Factor/bernoulli-numbers-3.factor Normal file
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[ 30 bernoulli-numbers ] time
...
Running time: 0.004331652 seconds

2
Task/Bernoulli-numbers/Fermat/bernoulli-numbers.fermat Normal file
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Func Bern(m) = Sigma<k=0,m>[Sigma<v=0,k>[(-1)^v*Bin(k,v)*(v+1)^m/(k+1)]].;
for i=0, 60 do b:=Bern(i); if b<>0 then !!(i,b) fi od;

54
Task/Bernoulli-numbers/FreeBASIC/bernoulli-numbers.basic Normal file
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' version 08-10-2016
' compile with: fbc -s console
' uses gmp
#Include Once "gmp.bi"
#Define max 60
Dim As Long n
Dim As ZString Ptr gmp_str :gmp_str = Allocate(1000) ' 1000 char
Dim Shared As Mpq_ptr tmp, big_j
tmp = Allocate(Len(__mpq_struct)) :Mpq_init(tmp)
big_j = Allocate(Len(__mpq_struct)) :Mpq_init(big_j)
Dim Shared As Mpq_ptr a(max), b(max)
For n = 0 To max
A(n) = Allocate(Len(__mpq_struct)) :Mpq_init(A(n))
B(n) = Allocate(Len(__mpq_struct)) :Mpq_init(B(n))
Next
Function Bernoulli(n As Integer) As Mpq_ptr
Dim As Long m, j
For m = 0 To n
Mpq_set_ui(A(m), 1, m + 1)
For j = m To 1 Step - 1
Mpq_sub(tmp, A(j - 1), A(j))
Mpq_set_ui(big_j, j, 1) 'big_j = j
Mpq_mul(A(j - 1), big_j, tmp)
Next
Next
Return A(0)
End Function
' ------=< MAIN >=------
For n = 0 To max
Mpq_set(B(n), Bernoulli(n))
Mpq_get_str(gmp_str, 10, B(n))
If *gmp_str <> "0" Then
If *gmp_str = "1" Then *gmp_str = "1/1"
Print Using "B(##) = "; n;
Print Space(45 - InStr(*gmp_str, "/")); *gmp_str
End If
Next
' empty keyboard buffer
While Inkey <> "" :Wend
Print :Print "hit any key to end program"
Sleep
End

25
Task/Bernoulli-numbers/Frink/bernoulli-numbers.frink Normal file
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BernoulliNumber[n] :=
{
a = new array
for m = 0 to n
{
a@m = 1/(m+1)
for j = m to 1 step -1
a@(j-1) = j * (a@(j-1) - a@j)
}
return a@0
}
result = new array
for n=0 to 60
{
b = BernoulliNumber[n]
if b != 0
{
[num,den] = numeratorDenominator[b]
result.push[[n, num, "/", den]]
}
}
println[formatTable[result, "right"]]

6
Task/Bernoulli-numbers/FunL/bernoulli-numbers.funl Normal file
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import integers.choose
def B( n ) = sum( 1/(k + 1)*sum((if 2|r then 1 else -1)*choose(k, r)*(r^n) | r <- 0..k) | k <- 0..n )
for i <- 0..60 if i == 1 or 2|i
printf( "B(%2d) = %s\n", i, B(i) )

36
Task/Bernoulli-numbers/GAP/bernoulli-numbers.gap Normal file
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for a in Filtered(List([0 .. 60], n -> [n, Bernoulli(n)]), x -> x[2] <> 0) do
Print(a, "\n");
od;
[ 0, 1 ]
[ 1, -1/2 ]
[ 2, 1/6 ]
[ 4, -1/30 ]
[ 6, 1/42 ]
[ 8, -1/30 ]
[ 10, 5/66 ]
[ 12, -691/2730 ]
[ 14, 7/6 ]
[ 16, -3617/510 ]
[ 18, 43867/798 ]
[ 20, -174611/330 ]
[ 22, 854513/138 ]
[ 24, -236364091/2730 ]
[ 26, 8553103/6 ]
[ 28, -23749461029/870 ]
[ 30, 8615841276005/14322 ]
[ 32, -7709321041217/510 ]
[ 34, 2577687858367/6 ]
[ 36, -26315271553053477373/1919190 ]
[ 38, 2929993913841559/6 ]
[ 40, -261082718496449122051/13530 ]
[ 42, 1520097643918070802691/1806 ]
[ 44, -27833269579301024235023/690 ]
[ 46, 596451111593912163277961/282 ]
[ 48, -5609403368997817686249127547/46410 ]
[ 50, 495057205241079648212477525/66 ]
[ 52, -801165718135489957347924991853/1590 ]
[ 54, 29149963634884862421418123812691/798 ]
[ 56, -2479392929313226753685415739663229/870 ]
[ 58, 84483613348880041862046775994036021/354 ]
[ 60, -1215233140483755572040304994079820246041491/56786730 ]

27
Task/Bernoulli-numbers/Go/bernoulli-numbers.go Normal file
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package main
import (
"fmt"
"math/big"
)
func b(n int) *big.Rat {
var f big.Rat
a := make([]big.Rat, n+1)
for m := range a {
a[m].SetFrac64(1, int64(m+1))
for j := m; j >= 1; j-- {
d := &a[j-1]
d.Mul(f.SetInt64(int64(j)), d.Sub(d, &a[j]))
}
}
return f.Set(&a[0])
}
func main() {
for n := 0; n <= 60; n++ {
if b := b(n); b.Num().BitLen() > 0 {
fmt.Printf("B(%2d) =%45s/%s\n", n, b.Num(), b.Denom())
}
}
}

23
Task/Bernoulli-numbers/Haskell/bernoulli-numbers-1.hs Normal file
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import Data.Ratio
import System.Environment
main = getArgs >>= printM . defaultArg
where
defaultArg as =
if null as
then 60
else read (head as)
printM m =
mapM_ (putStrLn . printP) .
takeWhile ((<= m) . fst) . filter (\(_, b) -> b /= 0 % 1) . zip [0 ..] $
bernoullis
printP (i, r) =
"B(" ++ show i ++ ") = " ++ show (numerator r) ++ "/" ++ show (denominator r)
bernoullis = map head . iterate (ulli 1) . map berno $ enumFrom 0
where
berno i = 1 % (i + 1)
ulli _ [_] = []
ulli i (x:y:xs) = (i % 1) * (x - y) : ulli (i + 1) (y : xs)

56
Task/Bernoulli-numbers/Haskell/bernoulli-numbers-2.hs Normal file
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import Data.Bool (bool)
import Data.Ratio (Ratio, denominator, numerator, (%))
-------------------- BERNOULLI NUMBERS -------------------
bernouillis :: Integer -> [Rational]
bernouillis =
fmap head
. tail
. scanl faulhaber []
. enumFromTo 0
faulhaber :: [Ratio Integer] -> Integer -> [Ratio Integer]
faulhaber rs n =
(:) =<< (-) 1 . sum $
zipWith ((*) . (n %)) [2 ..] rs
--------------------------- TEST -------------------------
main :: IO ()
main = do
let xs = bernouillis 60
w = length $ show (numerator (last xs))
putStrLn $
fTable
"Bernouillis from Faulhaber triangle:\n"
(show . fst)
(showRatio w . snd)
id
(filter ((0 /=) . snd) $ zip [0 ..] xs)
------------------------ FORMATTING ----------------------
fTable ::
String ->
(a -> String) ->
(b -> String) ->
(a -> b) ->
[a] ->
String
fTable s xShow fxShow f xs =
let w = maximum (length . xShow <$> xs)
in unlines $
s :
fmap
( ((<>) . rjust w ' ' . xShow)
<*> ((" -> " <>) . fxShow . f)
)
xs
showRatio :: Int -> Rational -> String
showRatio w r =
let d = denominator r
in rjust w ' ' $ show (numerator r)
<> bool [] (" / " <> show d) (1 /= d)
rjust :: Int -> a -> [a] -> [a]
rjust n c = drop . length <*> (replicate n c <>)

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Task/Bernoulli-numbers/Icon/bernoulli-numbers.icon Normal file
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link "rational"
procedure main(args)
limit := integer(!args) | 60
every b := bernoulli(i := 0 to limit) do
if b.numer > 0 then write(right(i,3),": ",align(rat2str(b),60))
end
procedure bernoulli(n)
(A := table(0))[0] := rational(1,1,1)
every m := 1 to n do {
A[m] := rational(1,m+1,1)
every j := m to 1 by -1 do A[j-1] := mpyrat(rational(j,1,1), subrat(A[j-1],A[j]))
}
return A[0]
end
procedure align(r,n)
return repl(" ",n-find("/",r))||r
end

1
Task/Bernoulli-numbers/J/bernoulli-numbers-1.j Normal file
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B=: {.&1 %. (i. ! ])@>:@i.@x:

33
Task/Bernoulli-numbers/J/bernoulli-numbers-2.j Normal file
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'B' ,. rplc&'r/_-'"1": (#~ 0 ~: {:"1)(i. ,. B) 61
B 0 1
B 1 -1/2
B 2 1/6
B 4 -1/30
B 6 1/42
B 8 -1/30
B10 5/66
B12 -691/2730
B14 7/6
B16 -3617/510
B18 43867/798
B20 -174611/330
B22 854513/138
B24 -236364091/2730
B26 8553103/6
B28 -23749461029/870
B30 8615841276005/14322
B32 -7709321041217/510
B34 2577687858367/6
B36 -26315271553053477373/1919190
B38 2929993913841559/6
B40 -261082718496449122051/13530
B42 1520097643918070802691/1806
B44 -27833269579301024235023/690
B46 596451111593912163277961/282
B48 -5609403368997817686249127547/46410
B50 495057205241079648212477525/66
B52 -801165718135489957347924991853/1590
B54 29149963634884862421418123812691/798
B56 -2479392929313226753685415739663229/870
B58 84483613348880041862046775994036021/354
B60 -1215233140483755572040304994079820246041491/56786730

22
Task/Bernoulli-numbers/Java/bernoulli-numbers.java Normal file
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import org.apache.commons.math3.fraction.BigFraction;
public class BernoulliNumbers {
public static void main(String[] args) {
for (int n = 0; n <= 60; n++) {
BigFraction b = bernouilli(n);
if (!b.equals(BigFraction.ZERO))
System.out.printf("B(%-2d) = %-1s%n", n , b);
}
}
static BigFraction bernouilli(int n) {
BigFraction[] A = new BigFraction[n + 1];
for (int m = 0; m <= n; m++) {
A[m] = new BigFraction(1, (m + 1));
for (int j = m; j >= 1; j--)
A[j - 1] = (A[j - 1].subtract(A[j])).multiply(new BigFraction(j));
}
return A[0];
}
}

39
Task/Bernoulli-numbers/Jq/bernoulli-numbers-1.jq Normal file
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# def negate:
# def lessOrEqual(x; y): # x <= y
# def long_add(x;y): # x+y
# def long_minus(x;y): # x-y
# def long_multiply(x;y) # x*y
# def long_divide(x;y): # x/y => [q,r]
# def long_div(x;y) # integer division
# def long_mod(x;y) # %
# In all cases, x and y must be strings
def negate: (- tonumber) | tostring;
def lessOrEqual(num1; num2): (num1|tonumber) <= (num2|tonumber);
def long_add(num1; num2): ((num1|tonumber) + (num2|tonumber)) | tostring;
def long_minus(x;y): ((num1|tonumber) - (num2|tonumber)) | tostring;
# multiply two decimal strings, which may be signed (+ or -)
def long_multiply(num1; num2):
((num1|tonumber) * (num2|tonumber)) | tostring;
# return [quotient, remainder]
# 0/0 = 1; n/0 => error
def long_divide(xx;yy): # x/y => [q,r] imples x == (y * q) + r
def ld(x;y):
def abs: if . < 0 then -. else . end;
(x|abs) as $x | (y|abs) as $y
| (if (x >= 0 and y > 0) or (x < 0 and y < 0) then 1 else -1 end) as $sign
| (if x >= 0 then 1 else -1 end) as $sx
| [$sign * ($x / $y | floor), $sx * ($x % $y)];
ld( xx|tonumber; yy|tonumber) | map(tostring);
def long_div(x;y):
long_divide(x;y) | .[0];
def long_mod(x;y):
((x|tonumber) % (y|tonumber)) | tostring;

54
Task/Bernoulli-numbers/Jq/bernoulli-numbers-2.jq Normal file
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# A fraction is represented by [numerator, denominator] in reduced form, with the sign on top
# a and b should be BigInt; return a BigInt
def gcd(a; b):
def long_abs: . as $in | if lessOrEqual("0"; $in) then $in else negate end;
# subfunction rgcd expects [a,b] as input
# i.e. a ~ .[0] and b ~ .[1]
def rgcd:
.[0] as $a | .[1] as $b
| if $b == "0" then $a
else [$b, long_mod($a ; $b ) ] | rgcd
end;
a as $a | b as $b
| [$a,$b] | rgcd | long_abs ;
def normalize:
.[0] as $p | .[1] as $q
| if $p == "0" then ["0", "1"]
elif lessOrEqual($q ; "0") then [ ($p|negate), ($q|negate)] | normalize
else gcd($p; $q) as $g
| [ long_div($p;$g), long_div($q;$g) ]
end ;
# a and b should be fractions expressed in the form [p, q]
def add(a; b):
a as $a | b as $b
| if $a[1] == "1" and $b[1] == "1" then [ long_add($a[0]; $b[0]) , "1"]
elif $a[1] == $b[1] then [ long_add( $a[0]; $b[0]), $a[1] ] | normalize
elif $a[0] == "0" then $b
elif $b[0] == "0" then $a
else [ long_add( long_multiply($a[0]; $b[1]) ; long_multiply($b[0]; $a[1])),
long_multiply($a[1]; $b[1]) ]
| normalize
end ;
# a and/or b may be BigInts, or [p,q] fractions
def multiply(a; b):
a as $a | b as $b
| if ($a|type) == "string" and ($b|type) == "string" then [ long_multiply($a; $b), "1"]
else
if $a|type == "string" then [ long_multiply( $a; $b[0]), $b[1] ]
elif $b|type == "string" then [ long_multiply( $b; $a[0]), $a[1] ]
else [ long_multiply( $a[0]; $b[0]), long_multiply($a[1]; $b[1]) ]
end
| normalize
end ;
def minus(a; b):
a as $a | b as $b
| if $a == $b then ["0", "1"]
else add($a; [ ($b[0]|negate), $b[1] ] )
end ;

10
Task/Bernoulli-numbers/Jq/bernoulli-numbers-3.jq Normal file
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# Using the algorithm in the task description:
def bernoulli(n):
reduce range(0; n+1) as $m
( [];
.[$m] = ["1", long_add($m|tostring; "1")] # i.e. 1 / ($m+1)
| reduce ($m - range(0 ; $m)) as $j
(.;
.[$j-1] = multiply( [($j|tostring), "1"]; minus( .[$j-1] ; .[$j]) ) ))
| .[0] # (which is Bn)
;

2
Task/Bernoulli-numbers/Jq/bernoulli-numbers-4.jq Normal file
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range(0;61)
| if . % 2 == 0 or . == 1 then "\(.): \(bernoulli(.) )" else empty end

33
Task/Bernoulli-numbers/Jq/bernoulli-numbers-5.jq Normal file
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$ jq -n -r -f Bernoulli.jq
0: ["1","1"]
1: ["1","2"]
2: ["1","6"]
4: ["-1","30"]
6: ["1","42"]
8: ["-1","30"]
10: ["5","66"]
12: ["-691","2730"]
14: ["7","6"]
16: ["-3617","510"]
18: ["43867","798"]
20: ["-174611","330"]
22: ["854513","138"]
24: ["-236364091","2730"]
26: ["8553103","6"]
28: ["-23749461029","870"]
30: ["8615841276005","14322"]
32: ["-7709321041217","510"]
34: ["2577687858367","6"]
36: ["-26315271553053477373","1919190"]
38: ["2929993913841559","6"]
40: ["-261082718496449122051","13530"]
42: ["1520097643918070802691","1806"]
44: ["-27833269579301024235023","690"]
46: ["596451111593912163277961","282"]
48: ["-5609403368997817686249127547","46410"]
50: ["495057205241079648212477525","66"]
52: ["-801165718135489957347924991853","1590"]
54: ["29149963634884862421418123812691","798"]
56: ["-2479392929313226753685415739663229","870"]
58: ["84483613348880041862046775994036021","354"]
60: ["-1215233140483755572040304994079820246041491","56786730"]

46
Task/Bernoulli-numbers/Julia/bernoulli-numbers.julia Normal file
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function bernoulli(n)
A = Vector{Rational{BigInt}}(undef, n + 1)
for m = 0 : n
A[m + 1] = 1 // (m + 1)
for j = m : -1 : 1
A[j] = j * (A[j] - A[j + 1])
end
end
return A[1]
end
function display(n)
B = map(bernoulli, 0 : n)
pad = mapreduce(x -> ndigits(numerator(x)) + Int(x < 0), max, B)
argdigits = ndigits(n)
for i = 0 : n
if numerator(B[i + 1]) & 1 == 1
println(
"B(", lpad(i, argdigits), ") = ",
lpad(numerator(B[i + 1]), pad), " / ", denominator(B[i + 1])
)
end
end
end
display(60)
# Alternative: Following the comment in the Perl section it is much more efficient
# to compute the list of numbers instead of one number after the other.
function BernoulliList(len)
A = Vector{Rational{BigInt}}(undef, len + 1)
B = similar(A)
for n in 0 : len
A[n + 1] = 1 // (n + 1)
for j = n : -1 : 1
A[j] = j * (A[j] - A[j + 1])
end
B[n + 1] = A[1]
end
return B
end
for (n, b) in enumerate(BernoulliList(60))
isodd(numerator(b)) && println("B($(n-1)) = $b")
end

22
Task/Bernoulli-numbers/Kotlin/bernoulli-numbers.kotlin Normal file
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import org.apache.commons.math3.fraction.BigFraction
object Bernoulli {
operator fun invoke(n: Int) : BigFraction {
val A = Array(n + 1, init)
for (m in 0..n)
for (j in m downTo 1)
A[j - 1] = A[j - 1].subtract(A[j]).multiply(integers[j])
return A.first()
}
val max = 60
private val init = { m: Int -> BigFraction(1, m + 1) }
private val integers = Array(max + 1, { m: Int -> BigFraction(m) } )
}
fun main(args: Array<String>) {
for (n in 0..Bernoulli.max)
if (n % 2 == 0 || n == 1)
System.out.printf("B(%-2d) = %-1s%n", n, Bernoulli(n))
}

45
Task/Bernoulli-numbers/Lua/bernoulli-numbers.lua Normal file
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#!/usr/bin/env luajit
local gmp = require 'gmp' ('libgmp')
local ffi = require'ffi'
local mpz, mpq = gmp.types.z, gmp.types.q
local function mpq_for(buf, op, n)
for i=0,n-1 do
op(buf[i])
end
end
local function bernoulli(rop, n)
local a=ffi.new("mpq_t[?]", n+1)
mpq_for(a, gmp.q_init, n+1)
for m=0,n do
gmp.q_set_ui(a[m],1, m+1)
for j=m,1,-1 do
gmp.q_sub(a[j-1], a[j], a[j-1])
gmp.q_set_ui(rop, j, 1)
gmp.q_mul(a[j-1], a[j-1], rop)
end
end
gmp.q_set(rop,a[0])
mpq_for(a, gmp.q_clear, n+1)
end
do --MAIN
local rop=mpq()
local n,d=mpz(),mpz()
gmp.q_init(rop)
gmp.z_inits(n, d)
local to=arg[1] and tonumber(arg[1]) or 60
local from=arg[2] and tonumber(arg[2]) or 0
if from~=0 then to,from=from,to end
for i=from,to do
bernoulli(rop, i)
if gmp.q_cmp_ui(rop, 0, 1)~=0 then
gmp.q_get_num(n, rop)
gmp.q_get_den(d, rop)
gmp.printf("B(%-2g) = %44Zd / %Zd\n", i, n, d)
end
end
gmp.z_clears(n,d)
gmp.q_clear(rop)
end

1
Task/Bernoulli-numbers/Maple/bernoulli-numbers.maple Normal file
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print(select(n->n[2]<>0,[seq([n,bernoulli(n,1)],n=0..60)]));

11
Task/Bernoulli-numbers/Mathematica/bernoulli-numbers-1.math Normal file
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bernoulli[n_] := Module[{a = ConstantArray[0, n + 2]},
Do[
a[[m]] = 1/m;
If[m == 1 && a[[1]] != 0, Print[{m - 1, a[[1]]}]];
Do[
a[[j - 1]] = (j - 1)*(a[[j - 1]] - a[[j]]);
If[j == 2 && a[[1]] != 0, Print[{m - 1, a[[1]]}]];
, {j, m, 2, -1}];
, {m, 1, n + 1}];
]
bernoulli[60]

2
Task/Bernoulli-numbers/Mathematica/bernoulli-numbers-2.math Normal file
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Table[{i, BernoulliB[i]}, {i, 0, 60}];
Select[%, #[[2]] != 0 &] // TableForm

42
Task/Bernoulli-numbers/Nim/bernoulli-numbers.nim Normal file
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import bignum
import strformat
const Lim = 60
#---------------------------------------------------------------------------------------------------
proc bernoulli(n: Natural): Rat =
## Compute a Bernoulli number using Akiyama–Tanigawa algorithm.
var a = newSeq[Rat](n + 1)
for m in 0..n:
a[m] = newRat(1, m + 1)
for j in countdown(m, 1):
a[j-1] = j * (a[j] - a[j-1])
result = a[0]
#———————————————————————————————————————————————————————————————————————————————————————————————————
type Info = tuple
n: int # Number index in Bernoulli sequence.
val: Rat # Bernoulli number.
var values: seq[Info] # List of values as Info tuples.
var maxLen = -1 # Maximum length.
# First step: compute the values and prepare for display.
for n in 0..Lim:
# Compute value.
if n != 1 and (n and 1) == 1: continue # Ignore odd "n" except 1.
let b = bernoulli(n)
# Check numerator length.
let len = ($b.num).len
if len > maxLen: maxLen = len
# Store information for next step.
values.add((n, b))
# Second step: display the values with '/' aligned.
for (n, b) in values:
let s = fmt"{($b.num).alignString(maxLen, '>')} / {b.denom}"
echo fmt"{n:2}: {s}"

1
Task/Bernoulli-numbers/PARI-GP/bernoulli-numbers.parigp Normal file
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for(n=0,60,t=bernfrac(n);if(t,print(n" "t)))

33
Task/Bernoulli-numbers/PL-I/bernoulli-numbers.pli Normal file
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Bern: procedure options (main); /* 4 July 2014 */
declare i fixed binary;
declare B complex fixed (31);
Bernoulli: procedure (n) returns (complex fixed (31));
declare n fixed binary;
declare anum(0:n) fixed (31), aden(0:n) fixed (31);
declare (j, m) fixed;
declare F fixed (31);
do m = 0 to n;
anum(m) = 1;
aden(m) = m+1;
do j = m to 1 by -1;
anum(j-1) = j*( aden(j)*anum(j-1) - aden(j-1)*anum(j) );
aden(j-1) = ( aden(j-1) * aden(j) );
F = gcd(abs(anum(j-1)), abs(aden(j-1)) );
if F ^= 1 then
do;
anum(j-1) = anum(j-1) / F;
aden(j-1) = aden(j-1) / F;
end;
end;
end;
return ( complex(anum(0), aden(0)) );
end Bernoulli;
do i = 0, 1, 2 to 36 by 2; /* 36 is upper limit imposed by hardware. */
B = Bernoulli(i);
put skip edit ('B(' , trim(i) , ')=' , real(B) , '/' , trim(imag(B)) )
(3 A, column(10), F(32), 2 A);
end;
end Bern;

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Task/Bernoulli-numbers/Pascal/bernoulli-numbers.pas Normal file
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(* Taken from the 'Ada 99' project, https://marquisdegeek.com/code_ada99 *)
program BernoulliForAda99;
uses BigDecimalMath; {library for arbitary high precision BCD numbers}
type
Fraction = object
private
numerator, denominator: BigDecimal;
public
procedure assign(n, d: Int64);
procedure subtract(rhs: Fraction);
procedure multiply(value: Int64);
procedure reduce();
procedure writeOutput();
end;
function gcd(a, b: BigDecimal):BigDecimal;
begin
if (b = 0) then begin
gcd := a;
end
else begin
gcd := gcd(b, a mod b);
end;
end;
procedure Fraction.writeOutput();
var sign : char;
begin
sign := ' ';
if (numerator<0) then sign := '-';
if (denominator<0) then sign := '-';
write(sign + BigDecimalToStr(abs(numerator)):45);
write(' / ');
write(BigDecimalToStr(abs(denominator)));
end;
procedure Fraction.assign(n, d: Int64);
begin
numerator := n;
denominator := d;
end;
procedure Fraction.subtract(rhs: Fraction);
begin
numerator := numerator * rhs.denominator;
numerator := numerator - (rhs.numerator * denominator);
denominator := denominator * rhs.denominator;
end;
procedure Fraction.multiply(value: Int64);
var
temp :BigDecimal;
begin
temp := value;
numerator := numerator * temp;
end;
procedure Fraction.reduce();
var gcdResult: BigDecimal;
begin
gcdResult := gcd(numerator, denominator);
begin
numerator := numerator div gcdResult; (* div is Int64 division *)
denominator := denominator div gcdResult; (* could also use round(d/r) *)
end;
end;
function calculateBernoulli(n: Int64) : Fraction;
var
m, j: Int64;
results: array of Fraction;
begin
setlength(results, 60) ; {largest value 60}
for m:= 0 to n do
begin
results[m].assign(1, m+1);
for j:= m downto 1 do
begin
results[j-1].subtract(results[j]);
results[j-1].multiply(j);
results[j-1].reduce();
end;
end;
calculateBernoulli := results[0];
end;
(* Main program starts here *)
var
b: Int64;
result: Fraction;
begin
writeln('Calculating Bernoulli numbers...');
writeln('B( 0) : 1 / 1');
for b:= 1 to 60 do
begin
if (b<3) or ((b mod 2) = 0) then begin
result := calculateBernoulli(b);
write('B(',b:2,')');
write(' : ');
result.writeOutput();
writeln;
end;
end;
end.

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Task/Bernoulli-numbers/Perl/bernoulli-numbers-1.pl Normal file
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#!perl
use strict;
use warnings;
use List::Util qw(max);
use Math::BigRat;
my $one = Math::BigRat->new(1);
sub bernoulli_print {
my @a;
for my $m ( 0 .. 60 ) {
push @a, $one / ($m + 1);
for my $j ( reverse 1 .. $m ) {
# This line:
( $a[$j-1] -= $a[$j] ) *= $j;
# is a faster version of the following line:
# $a[$j-1] = $j * ($a[$j-1] - $a[$j]);
# since it avoids unnecessary object creation.
}
next unless $a[0];
printf "B(%2d) = %44s/%s\n", $m, $a[0]->parts;
}
}
bernoulli_print();

6
Task/Bernoulli-numbers/Perl/bernoulli-numbers-2.pl Normal file
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use ntheory qw/bernfrac/;
for my $n (0 .. 60) {
my($num,$den) = bernfrac($n);
printf "B(%2d) = %44s/%s\n", $n, $num, $den if $num != 0;
}

6
Task/Bernoulli-numbers/Perl/bernoulli-numbers-3.pl Normal file
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use Math::Pari qw/bernfrac/;
for my $n (0 .. 60) {
my($num,$den) = split "/", bernfrac($n);
printf("B(%2d) = %44s/%s\n", $n, $num, $den||1) if $num != 0;
}

33
Task/Bernoulli-numbers/Phix/bernoulli-numbers.phix Normal file
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(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #000000;">builtins</span><span style="color: #0000FF;">/</span><span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">bernoulli</span><span style="color: #0000FF;">(</span><span style="color: #004080;">mpq</span> <span style="color: #000000;">rop</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpq_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpq_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">m</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">m</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">])</span>
<span style="color: #7060A8;">mpq_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rop</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">rop</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">mpq_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rop</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpq_free</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #004080;">mpq</span> <span style="color: #000000;">rop</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpq_init</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(),</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">60</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">bernoulli</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rop</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpq_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rop</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">mpq_get_num</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">rop</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpq_get_den</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">rop</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">string</span> <span style="color: #000000;">ns</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">ds</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"B(%2d) = %44s / %s\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ns</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ds</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_free</span><span style="color: #0000FF;">({</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">})</span>
<span style="color: #000000;">rop</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpq_free</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rop</span><span style="color: #0000FF;">)</span>
<!--

74
Task/Bernoulli-numbers/PicoLisp/bernoulli-numbers.l Normal file
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(load "@lib/frac.l")
(de fact (N)
(cache '(NIL) N
(if (=0 N) 1 (apply * (range 1 N))) ) )
(de binomial (N K)
(frac
(/
(fact N)
(* (fact (- N K)) (fact K)) )
1 ) )
(de A (N M)
(let Sum (0 . 1)
(for X M
(setq Sum
(f+
Sum
(f*
(binomial (+ N 3) (- N (* X 6)))
(berno (- N (* X 6)) ) ) ) ) )
Sum ) )
(de berno (N)
(cache '(NIL) N
(cond
((=0 N) (1 . 1))
((= 1 N) (-1 . 2))
((bit? 1 N) (0 . 1))
(T
(case (% N 6)
(0
(f/
(f-
(frac (+ N 3) 3)
(A N (/ N 6)) )
(binomial (+ N 3) N) ) )
(2
(f/
(f-
(frac (+ N 3) 3)
(A N (/ (- N 2) 6)) )
(binomial (+ N 3) N) ) )
(4
(f/
(f-
(f* (-1 . 1) (frac (+ N 3) 6))
(A N (/ (- N 4) 6)) )
(binomial (+ N 3) N) ) ) ) ) ) ) )
(de berno-brute (N)
(cache '(NIL) N
(let Sum (0 . 1)
(cond
((=0 N) (1 . 1))
((= 1 N) (-1 . 2))
((bit? 1 N) (0 . 1))
(T
(for (X 0 (> N X) (inc X))
(setq Sum
(f+
Sum
(f* (binomial (inc N) X) (berno-brute X)) ) ) )
(f/ (f* (-1 . 1) Sum) (binomial (inc N) N)) ) ) ) ) )
(for (N 0 (> 62 N) (inc N))
(if (or (= N 1) (not (bit? 1 N)))
(tab (2 4 -60) N " => " (sym (berno N))) ) )
(for (N 0 (> 400 N) (inc N))
(test (berno N) (berno-brute N)) )
(bye)

15
Task/Bernoulli-numbers/Python/bernoulli-numbers-1.py Normal file
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from fractions import Fraction as Fr
def bernoulli(n):
A = [0] * (n+1)
for m in range(n+1):
A[m] = Fr(1, m+1)
for j in range(m, 0, -1):
A[j-1] = j*(A[j-1] - A[j])
return A[0] # (which is Bn)
bn = [(i, bernoulli(i)) for i in range(61)]
bn = [(i, b) for i,b in bn if b]
width = max(len(str(b.numerator)) for i,b in bn)
for i,b in bn:
print('B(%2i) = %*i/%i' % (i, width, b.numerator, b.denominator))

14
Task/Bernoulli-numbers/Python/bernoulli-numbers-2.py Normal file
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def bernoulli2():
A, m = [], 0
while True:
A.append(Fr(1, m+1))
for j in range(m, 0, -1):
A[j-1] = j*(A[j-1] - A[j])
yield A[0] # (which is Bm)
m += 1
bn2 = [ix for ix in zip(range(61), bernoulli2())]
bn2 = [(i, b) for i,b in bn2 if b]
width = max(len(str(b.numerator)) for i,b in bn2)
for i,b in bn2:
print('B(%2i) = %*i/%i' % (i, width, b.numerator, b.denominator))

25
Task/Bernoulli-numbers/Quackery/bernoulli-numbers.quackery Normal file
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$ "bigrat.qky" loadfile
[ 1+
' [ [] ] over of swap
times
[ i^ 1+ n->v 1/v
join swap i^ poke
i^ times
[ dup i 1+ peek do
dip over swap i peek do
v- i 1+ n->v v*
join swap i poke ] ]
1 split drop do ] is bernoulli ( n --> n/d )
61 times
[ i^ bernoulli
2dup v0= iff
2drop
else
[ i^ 10 < if sp
i^ echo sp
vulgar$
char / over find
44 swap - times sp
echo$ cr ] ]

8
Task/Bernoulli-numbers/R/bernoulli-numbers.r Normal file
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library(pracma)
for (idx in c(1,2*0:30)) {
b <- bernoulli(idx)
d <- as.character(denominator(b))
n <- as.character(numerator(b))
cat("B(",idx,") = ",n,"/",d,"\n", sep = "")
}

55
Task/Bernoulli-numbers/REXX/bernoulli-numbers.rexx Normal file
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/*REXX program calculates N number of Bernoulli numbers expressed as vulgar fractions.*/
parse arg N .; if N=='' | N=="," then N= 60 /*Not specified? Then use the default.*/
numeric digits max(9, n*2) /*increase the decimal digits if needed*/
w= max(length(N), 4); Nw= N + w + N % 4 /*used for aligning (output) fractions.*/
say 'B(n)' center("Bernoulli numbers expressed as vulgar fractions", max(78-w, Nw) )
say copies('─',w) copies("─", max(78-w,Nw+2*w)) /*display 2nd line of title, separators*/
!.= .; do #=0 to N /*process the numbers from 0 ──► N. */
b= bern(#); if b==0 then iterate /*calculate Bernoulli number, skip if 0*/
indent= max(0, nW - pos('/', b) ) /*calculate the alignment (indentation)*/
say right(#, w) left('', indent) b /*display the indented Bernoulli number*/
end /*#*/ /* [↑] align the Bernoulli fractions. */
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
bern: parse arg x; if x==0 then return '1/1' /*handle the special case of zero. */
if x==1 then return '-1/2' /* " " " " " one. */
if x//2 then return 0 /* " " " " " odds > 1.*/
do j=2 to x by 2; jp= j+1 /*process the positive integers up to X*/
sn= 1 - j /*define the numerator. */
sd= 2 /* " " denominator. */
do k=2 to j-1 by 2 /*calculate a SN/SD sequence. */
parse var @.k bn '/' ad /*get a previously calculated fraction.*/
an= comb(jp, k) * bn /*use COMBination for the next term. */
$lcm= LCM(sd, ad) /*use Least Common Denominator function*/
sn= $lcm % sd * sn; sd= $lcm /*calculate the current numerator. */
an= $lcm % ad * an /* " " next " */
sn= sn + an /* " " current " */
end /*k*/ /* [↑] calculate the SN/SD sequence.*/
sn= -sn /*flip the sign for the numerator. */
sd= sd * jp /*calculate the denominator. */
if sn\==1 then do; _= GCD(sn, sd) /*get the Greatest Common Denominator.*/
sn= sn%_; sd= sd%_ /*reduce the numerator and denominator.*/
end /* [↑] done with the reduction(s). */
@.j= sn'/'sd /*save the result for the next round. */
end /*j*/ /* [↑] done calculating Bernoulli #'s.*/
return sn'/'sd
/*──────────────────────────────────────────────────────────────────────────────────────*/
comb: procedure expose !.; parse arg x,y; if x==y then return 1
if !.C.x.y\==. then return !.C.x.y /*combination computed before?*/
if x-y < y then y= x-y /*x-y < y? Then use a new Y.*/
z= perm(x, y); do j=2 for y-1; z= z % j
end /*j*/
!.C.x.y= z; return z /*assign memoization & return.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
GCD: procedure; parse arg x,y; x= abs(x)
do until y==0; parse value x//y y with y x; end; return x
/*──────────────────────────────────────────────────────────────────────────────────────*/
LCM: procedure; parse arg x,y /*X=ABS(X); Y=ABS(Y) not needed for Bernoulli #s.*/
/*IF Y==0 THEN RETURN 0 " " " " " */
$= x * y /*calculate part of the LCM here. */
do until y==0; parse value x//y y with y x
end /*until*/ /* [↑] this is a short & fast GCD*/
return $ % x /*divide the pre─calculated value.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
perm: procedure expose !.; parse arg x,y; if !.P.x.y\==. then return !.P.x.y
z= 1; do j=x-y+1 to x; z= z*j; end; !.P.x.y= z; return z

84
Task/Bernoulli-numbers/Racket/bernoulli-numbers.rkt Normal file
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#lang racket
;; For: http://rosettacode.org/wiki/Bernoulli_numbers
;; As described in task...
(define (bernoulli.1 n)
(define A (make-vector (add1 n)))
(for ((m (in-range 0 (add1 n))))
(vector-set! A m (/ (add1 m)))
(for ((j (in-range m (sub1 1) -1)))
(define new-A_j-1 (* j (- (vector-ref A (sub1 j)) (vector-ref A j))))
(vector-set! A (sub1 j) new-A_j-1)))
(vector-ref A 0))
(define (non-zero-bernoulli-indices s)
(sequence-filter (λ (n) (or (even? n) (= n 1))) s))
(define (bernoulli_0..n B N)
(for/list ((n (non-zero-bernoulli-indices (in-range (add1 N))))) (B n)))
;; From REXX description / http://mathworld.wolfram.com/BernoulliNumber.html #33
;; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
;; bernoulli.2 is for illustrative purposes, binomial is very costly if there is no memoisation
;; (which math/number-theory doesn't do)
(require (only-in math/number-theory binomial))
(define (bernoulli.2 n)
(for/sum ((k (in-range 0 (add1 n))))
(* (/ (add1 k))
(for/sum ((r (in-range 0 (add1 k))))
(* (expt -1 r) (binomial k r) (expt r n))))))
;; Three things to do:
;; 1. (expt -1 r): is 1 for even r, -1 for odd r... split the sum between those two.
;; 2. splitting the sum might has arithmetic advantages, too. We're using rationals, so the smaller
;; summations should require less normalisation of intermediate, fractional results
;; 3. a memoised binomial... although the one from math/number-theory is fast, it is (and its
;; factorials are) computed every time which is redundant
(define kCr-memo (make-hasheq))
(define !-memo (make-vector 1000 #f))
(vector-set! !-memo 0 1) ;; seed the memo
(define (! k)
(cond [(vector-ref !-memo k) => values]
[else (define k! (* k (! (- k 1)))) (vector-set! !-memo k k!) k!]))
(define (kCr k r)
; If we want (kCr ... r>1000000) we'll have to reconsider this. However, until then...
(define hash-key (+ (* 1000000 k) r))
(hash-ref! kCr-memo hash-key (λ () (/ (! k) (! r) (! (- k r))))))
(define (bernoulli.3 n)
(for/sum ((k (in-range 0 (add1 n))))
(define k+1 (add1 k))
(* (/ k+1)
(- (for/sum ((r (in-range 0 k+1 2))) (* (kCr k r) (expt r n)))
(for/sum ((r (in-range 1 k+1 2))) (* (kCr k r) (expt r n)))))))
(define (display/align-fractions caption/idx-fmt Bs)
;; widths are one more than the order of magnitude
(define oom+1 (compose add1 order-of-magnitude))
(define-values (I-width N-width D-width)
(for/fold ((I 0) (N 0) (D 0))
((b Bs) (n (non-zero-bernoulli-indices (in-naturals))))
(define +b (abs b))
(values (max I (oom+1 (max n 1)))
(max N (+ (oom+1 (numerator +b)) (if (negative? b) 1 0)))
(max D (oom+1 (denominator +b))))))
(define (~a/w/a n w a) (~a n #:width w #:align a))
(for ((n (non-zero-bernoulli-indices (in-naturals))) (b Bs))
(printf "~a ~a/~a~%"
(format caption/idx-fmt (~a/w/a n I-width 'right))
(~a/w/a (numerator b) N-width 'right)
(~a/w/a (denominator b) D-width 'left))))
(module+ main
(display/align-fractions "B(~a) =" (bernoulli_0..n bernoulli.3 60)))
(module+ test
(require rackunit)
; correctness and timing tests
(check-match (time (bernoulli_0..n bernoulli.1 60))
(list 1/1 (app abs 1/2) 1/6 -1/30 1/42 -1/30 _ ...))
(check-match (time (bernoulli_0..n bernoulli.2 60))
(list 1/1 (app abs 1/2) 1/6 -1/30 1/42 -1/30 _ ...))
(check-match (time (bernoulli_0..n bernoulli.3 60))
(list 1/1 (app abs 1/2) 1/6 -1/30 1/42 -1/30 _ ...))
; timing only ...
(void (time (bernoulli_0..n bernoulli.3 100))))

17
Task/Bernoulli-numbers/Raku/bernoulli-numbers-1.raku Normal file
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sub bernoulli($n) {
my @a;
for 0..$n -> $m {
@a[$m] = FatRat.new(1, $m + 1);
for reverse 1..$m -> $j {
@a[$j - 1] = $j * (@a[$j - 1] - @a[$j]);
}
}
return @a[0];
}
constant @bpairs = grep *.value.so, ($_ => bernoulli($_) for 0..60);
my $width = max @bpairs.map: *.value.numerator.chars;
my $form = "B(%2d) = \%{$width}d/%d\n";
printf $form, .key, .value.nude for @bpairs;

18
Task/Bernoulli-numbers/Raku/bernoulli-numbers-2.raku Normal file
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constant bernoulli = gather {
my @a;
for 0..* -> $m {
@a = FatRat.new(1, $m + 1),
-> $prev {
my $j = @a.elems;
$j * (@a.shift - $prev);
} ... { not @a.elems }
take $m => @a[*-1] if @a[*-1];
}
}
constant @bpairs = bernoulli[^52];
my $width = max @bpairs.map: *.value.numerator.chars;
my $form = "B(%d)\t= \%{$width}d/%d\n";
printf $form, .key, .value.nude for @bpairs;

23
Task/Bernoulli-numbers/Raku/bernoulli-numbers-3.raku Normal file
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sub infix:<bop>(\prev, \this) {
this.key => this.key * (this.value - prev.value)
}
sub next-bernoulli ( (:key($pm), :value(@pa)) ) {
$pm + 1 => [
map *.value,
[\bop] ($pm + 2 ... 1) Z=> FatRat.new(1, $pm + 2), |@pa
]
}
constant bernoulli =
grep *.value,
map { .key => .value[*-1] },
(0 => [FatRat.new(1,1)], &next-bernoulli ... *)
;
constant @bpairs = bernoulli[^52];
my $width = max @bpairs.map: *.value.numerator.chars;
my $form = "B(%d)\t= \%{$width}d/%d\n";
printf $form, .key, .value.nude for @bpairs;

12
Task/Bernoulli-numbers/Ruby/bernoulli-numbers.rb Normal file
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bernoulli = Enumerator.new do |y|
ar = []
0.step do |m|
ar << Rational(1, m+1)
m.downto(1){|j| ar[j-1] = j*(ar[j-1] - ar[j]) }
y << ar.first # yield
end
end
b_nums = bernoulli.take(61)
width = b_nums.map{|b| b.numerator.to_s.size}.max
b_nums.each_with_index {|b,i| puts "B(%2i) = %*i/%i" % [i, width, b.numerator, b.denominator] unless b.zero? }

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Task/Bernoulli-numbers/Rust/bernoulli-numbers.rust Normal file
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// 2.5 implementations presented here: naive, optimized, and an iterator using
// the optimized function. The speeds vary significantly: relative
// speeds of optimized:iterator:naive implementations is 625:25:1.
#![feature(test)]
extern crate num;
extern crate test;
use num::bigint::{BigInt, ToBigInt};
use num::rational::{BigRational};
use std::cmp::max;
use std::env;
use std::ops::{Mul, Sub};
use std::process;
struct Bn {
value: BigRational,
index: i32
}
struct Context {
bigone_const: BigInt,
a: Vec<BigRational>,
index: i32 // Counter for iterator implementation
}
impl Context {
pub fn new() -> Context {
let bigone = 1.to_bigint().unwrap();
let a_vec: Vec<BigRational> = vec![];
Context {
bigone_const: bigone,
a: a_vec,
index: -1
}
}
}
impl Iterator for Context {
type Item = Bn;
fn next(&mut self) -> Option<Bn> {
self.index += 1;
Some(Bn { value: bernoulli(self.index as usize, self), index: self.index })
}
}
fn help() {
println!("Usage: bernoulli_numbers <up_to>");
}
fn main() {
let args: Vec<String> = env::args().collect();
let mut up_to: usize = 60;
match args.len() {
1 => {},
2 => {
up_to = args[1].parse::<usize>().unwrap();
},
_ => {
help();
process::exit(0);
}
}
let context = Context::new();
// Collect the solutions by using the Context iterator
// (this is not as fast as calling the optimized function directly).
let res = context.take(up_to + 1).collect::<Vec<_>>();
let width = res.iter().fold(0, |a, r| max(a, r.value.numer().to_string().len()));
for r in res.iter().filter(|r| *r.value.numer() != ToBigInt::to_bigint(&0).unwrap()) {
println!("B({:>2}) = {:>2$} / {denom}", r.index, r.value.numer(), width,
denom = r.value.denom());
}
}
// Implementation with no reused calculations.
fn _bernoulli_naive(n: usize, c: &mut Context) -> BigRational {
for m in 0..n + 1 {
c.a.push(BigRational::new(c.bigone_const.clone(), (m + 1).to_bigint().unwrap()));
for j in (1..m + 1).rev() {
c.a[j - 1] = (c.a[j - 1].clone().sub(c.a[j].clone())).mul(
BigRational::new(j.to_bigint().unwrap(), c.bigone_const.clone())
);
}
}
c.a[0].reduced()
}
// Implementation with reused calculations (does not require sequential calls).
fn bernoulli(n: usize, c: &mut Context) -> BigRational {
for i in 0..n + 1 {
if i >= c.a.len() {
c.a.push(BigRational::new(c.bigone_const.clone(), (i + 1).to_bigint().unwrap()));
for j in (1..i + 1).rev() {
c.a[j - 1] = (c.a[j - 1].clone().sub(c.a[j].clone())).mul(
BigRational::new(j.to_bigint().unwrap(), c.bigone_const.clone())
);
}
}
}
c.a[0].reduced()
}
#[cfg(test)]
mod tests {
use super::{Bn, Context, bernoulli, _bernoulli_naive};
use num::rational::{BigRational};
use std::str::FromStr;
use test::Bencher;
// [tests elided]
#[bench]
fn bench_bernoulli_naive(b: &mut Bencher) {
let mut context = Context::new();
b.iter(|| {
let mut res: Vec<Bn> = vec![];
for n in 0..30 + 1 {
let b = _bernoulli_naive(n, &mut context);
res.push(Bn { value:b.clone(), index: n as i32});
}
});
}
#[bench]
fn bench_bernoulli(b: &mut Bencher) {
let mut context = Context::new();
b.iter(|| {
let mut res: Vec<Bn> = vec![];
for n in 0..30 + 1 {
let b = bernoulli(n, &mut context);
res.push(Bn { value:b.clone(), index: n as i32});
}
});
}
#[bench]
fn bench_bernoulli_iter(b: &mut Bencher) {
b.iter(|| {
let context = Context::new();
let _res = context.take(30 + 1).collect::<Vec<_>>();
});
}
}

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Task/Bernoulli-numbers/SPAD/bernoulli-numbers.spad Normal file
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for n in 0..60 | (b:=bernoulli(n)$INTHEORY; b~=0) repeat print [n,b]

49
Task/Bernoulli-numbers/Scala/bernoulli-numbers.scala Normal file
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/** Roll our own pared-down BigFraction class just for these Bernoulli Numbers */
case class BFraction( numerator:BigInt, denominator:BigInt ) {
require( denominator != BigInt(0), "Denominator cannot be zero" )
val gcd = numerator.gcd(denominator)
val num = numerator / gcd
val den = denominator / gcd
def unary_- = BFraction(-num, den)
def -( that:BFraction ) = that match {
case f if f.num == BigInt(0) => this
case f if f.den == this.den => BFraction(this.num - f.num, this.den)
case f => BFraction(((this.num * f.den) - (f.num * this.den)), this.den * f.den )
}
def *( that:Int ) = BFraction( num * that, den )
override def toString = num + " / " + den
}
def bernoulliB( n:Int ) : BFraction = {
val aa : Array[BFraction] = Array.ofDim(n+1)
for( m <- 0 to n ) {
aa(m) = BFraction(1,(m+1))
for( n <- m to 1 by -1 ) {
aa(n-1) = (aa(n-1) - aa(n)) * n
}
}
aa(0)
}
assert( {val b12 = bernoulliB(12); b12.num == -691 && b12.den == 2730 } )
val r = for( n <- 0 to 60; b = bernoulliB(n) if b.num != 0 ) yield (n, b)
val numeratorSize = r.map(_._2.num.toString.length).max
// Print the results
r foreach{ case (i,b) => {
val label = f"b($i)"
val num = (" " * (numeratorSize - b.num.toString.length)) + b.num
println( f"$label%-6s $num / ${b.den}" )
}}

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Task/Bernoulli-numbers/Scheme/bernoulli-numbers.ss Normal file
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; Return the n'th Bernoulli number.
(define bernoulli
(lambda (n)
(let ((a (make-vector (1+ n))))
(do ((m 0 (1+ m)))
((> m n))
(vector-set! a m (/ 1 (1+ m)))
(do ((j m (1- j)))
((< j 1))
(vector-set! a (1- j) (* j (- (vector-ref a (1- j)) (vector-ref a j))))))
(vector-ref a 0))))
; Convert a rational to a string. If an integer, ends with "/1".
(define rational->string
(lambda (rational)
(format "~a/~a" (numerator rational) (denominator rational))))
; Returns the string length of the numerator of a rational.
(define rational-numerator-length
(lambda (rational)
(string-length (format "~a" (numerator rational)))))
; Formats a rational with left-padding such that total length to the slash is as given.
(define rational-padded
(lambda (rational total-length-to-slash)
(let* ((length-padding (- total-length-to-slash (rational-numerator-length rational)))
(padding-string (make-string length-padding #\ )))
(string-append padding-string (rational->string rational)))))
; Return the Bernoulli numbers 0 through n in a list.
(define make-bernoulli-list
(lambda (n)
(if (= n 0)
(list (bernoulli n))
(append (make-bernoulli-list (1- n)) (list (bernoulli n))))))
; Print the non-zero Bernoulli numbers 0 through 60 aligning the slashes.
(let* ((bernoullis-list (make-bernoulli-list 60))
(numerator-lengths (map rational-numerator-length bernoullis-list))
(max-numerator-length (apply max numerator-lengths)))
(let print-bernoulli ((index 0) (numbers bernoullis-list))
(cond
((null? numbers))
((= 0 (car numbers))
(print-bernoulli (1+ index) (cdr numbers)))
(else
(printf "B(~2@a) = ~a~%" index (rational-padded (car numbers) max-numerator-length))
(print-bernoulli (1+ index) (cdr numbers))))))

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Task/Bernoulli-numbers/Seed7/bernoulli-numbers.seed7 Normal file
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$ include "seed7_05.s7i";
include "bigrat.s7i";
const func bigRational: bernoulli (in integer: n) is func
result
var bigRational: bernoulli is bigRational.value;
local
var integer: m is 0;
var integer: j is 0;
var array bigRational: a is 0 times bigRational.value;
begin
a := [0 .. n] times bigRational.value;
for m range 0 to n do
a[m] := 1_ / bigInteger(succ(m));
for j range m downto 1 do
a[pred(j)] := bigRational(j) * (a[j] - a[pred(j)]);
end for;
end for;
bernoulli := a[0];
end func;
const proc: main is func
local
var bigRational: bernoulli is bigRational.value;
var integer: i is 0;
begin
for i range 0 to 60 do
bernoulli := bernoulli(i);
if bernoulli <> bigRational.value then
writeln("B(" <& i lpad 2 <& ") = " <& bernoulli.numerator lpad 44 <&
" / " <& bernoulli.denominator rpad 8 <& " " <& bernoulli);
end if;
end for;
end func;

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Task/Bernoulli-numbers/Sidef/bernoulli-numbers-1.sidef Normal file
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say bernoulli(42).as_frac #=> 1520097643918070802691/1806

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Task/Bernoulli-numbers/Sidef/bernoulli-numbers-2.sidef Normal file
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func bernoulli_number(n) is cached {
n.is_one && return 1/2
n.is_odd && return 0
1 - sum(^n, {|k|
binomial(n,k) * __FUNC__(k) / (n - k + 1)
})
}
for n in (0..60) {
var Bn = bernoulli_number(n) || next
printf("B(%2d) = %44s / %s\n", n, Bn.nude)
}

9
Task/Bernoulli-numbers/Sidef/bernoulli-numbers-3.sidef Normal file
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func ramanujan_bernoulli_number(n) is cached {
return 1/2 if n.is_one
return 0 if n.is_odd
((n%6 == 4 ? -1/2 : 1) * (n+3)/3 - sum(1 .. (n - n%6)/6, {|k|
binomial(n+3, n - 6*k) * __FUNC__(n - 6*k)
})) / binomial(n+3, n)
}

15
Task/Bernoulli-numbers/Sidef/bernoulli-numbers-4.sidef Normal file
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func bernoulli_number_from_zeta(n) {
n.is_zero && return 1
n.is_one && return 1/2
n.is_odd && return 0
var log2B = (log(4*Num.tau*n)/2 + n*log(n) - n*log(Num.tau) - n)/log(2)
local Num!PREC = *(int(n + log2B) + (n <= 90 ? 18 : 0))
var K = 2*(n! / Num.tau**n)
var d = n.divisors.grep {|k| is_prime(k+1) }.prod {|k| k+1 }
var z = ceil((K*d).root(n-1)).primes.prod {|p| 1 - p.float**(-n) }
(-1)**(n/2 + 1) * int(ceil(d*K / z)) / d
}

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Task/Bernoulli-numbers/Sidef/bernoulli-numbers-5.sidef Normal file
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func bernoulli_print {
var a = []
for m in (0..60) {
a << 1/(m+1)
for j in (1..m -> flip) {
(a[j-1] -= a[j]) *= j
}
a[0] || next
printf("B(%2d) = %44s / %s\n", m, a[0].nude)
}
}
 
bernoulli_print()

29
Task/Bernoulli-numbers/Swift/bernoulli-numbers.swift Normal file
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import BigInt
public func bernoulli<T: BinaryInteger & SignedNumeric>(n: Int) -> Frac<T> {
guard n != 0 else {
return 1
}
var arr = [Frac<T>]()
for m in 0...n {
arr.append(Frac(numerator: 1, denominator: T(m) + 1))
for j in stride(from: m, through: 1, by: -1) {
arr[j-1] = (arr[j-1] - arr[j]) * Frac(numerator: T(j), denominator: 1)
}
}
return arr[0]
}
for n in 0...60 {
let b = bernoulli(n: n) as Frac<BigInt>
guard b != 0 else {
continue
}
print("B(\(n)) = \(b)")
}

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Task/Bernoulli-numbers/Tcl/bernoulli-numbers.tcl Normal file
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proc bernoulli {n} {
for {set m 0} {$m <= $n} {incr m} {
lappend A [list 1 [expr {$m + 1}]]
for {set j $m} {[set i $j] >= 1} {} {
lassign [lindex $A [incr j -1]] a1 b1
lassign [lindex $A $i] a2 b2
set x [set p [expr {$i * ($a1*$b2 - $a2*$b1)}]]
set y [set q [expr {$b1 * $b2}]]
while {$q} {set q [expr {$p % [set p $q]}]}
lset A $j [list [expr {$x/$p}] [expr {$y/$p}]]
}
}
return [lindex $A 0]
}
set len 0
for {set n 0} {$n <= 60} {incr n} {
set b [bernoulli $n]
if {[lindex $b 0]} {
lappend result $n {*}$b
set len [expr {max($len, [string length [lindex $b 0]])}]
}
}
foreach {n num denom} $result {
puts [format {B_%-2d = %*lld/%lld} $n $len $num $denom]
}

50
Task/Bernoulli-numbers/Visual-Basic-.NET/bernoulli-numbers.vb Normal file
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' Bernoulli numbers - vb.net - 06/03/2017
Imports System.Numerics 'BigInteger
Module Bernoulli_numbers
Function gcd_BigInt(ByVal x As BigInteger, ByVal y As BigInteger) As BigInteger
Dim y2 As BigInteger
x = BigInteger.Abs(x)
Do
y2 = BigInteger.Remainder(x, y)
x = y
y = y2
Loop Until y = 0
Return x
End Function 'gcd_BigInt
Sub bernoul_BigInt(n As Integer, ByRef bnum As BigInteger, ByRef bden As BigInteger)
Dim j, m As Integer
Dim f As BigInteger
Dim anum(), aden() As BigInteger
ReDim anum(n + 1), aden(n + 1)
For m = 0 To n
anum(m + 1) = 1
aden(m + 1) = m + 1
For j = m To 1 Step -1
anum(j) = j * (aden(j + 1) * anum(j) - aden(j) * anum(j + 1))
aden(j) = aden(j) * aden(j + 1)
f = gcd_BigInt(BigInteger.Abs(anum(j)), BigInteger.Abs(aden(j)))
If f <> 1 Then
anum(j) = anum(j) / f
aden(j) = aden(j) / f
End If
Next
Next
bnum = anum(1) : bden = aden(1)
End Sub 'bernoul_BigInt
Sub bernoulli_BigInt()
Dim i As Integer
Dim bnum, bden As BigInteger
bnum = 0 : bden = 0
For i = 0 To 60
bernoul_BigInt(i, bnum, bden)
If bnum <> 0 Then
Console.WriteLine("B(" & i & ")=" & bnum.ToString("D") & "/" & bden.ToString("D"))
End If
Next i
End Sub 'bernoulli_BigInt
End Module 'Bernoulli_numbers

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Task/Bernoulli-numbers/Wren/bernoulli-numbers.wren Normal file
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import "/fmt" for Fmt
import "/big" for BigRat
var bernoulli = Fn.new { |n|
if (n < 0) Fiber.abort("Argument must be non-negative")
var a = List.filled(n+1, null)
for (m in 0..n) {
a[m] = BigRat.new(1, m+1)
var j = m
while (j >= 1) {
a[j-1] = (a[j-1] - a[j]) * BigRat.new(j, 1)
j = j - 1
}
}
return (n != 1) ? a[0] : -a[0] // 'first' Bernoulli number
}
for (n in 0..60) {
var b = bernoulli.call(n)
if (b != BigRat.zero) Fmt.print("B($2d) = $44i / $i", n, b.num, b.den)
}

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Task/Bernoulli-numbers/Zkl/bernoulli-numbers-1.zkl Normal file
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class Rational{ // Weenie Rational class, can handle BigInts
fcn init(_a,_b){ var a=_a, b=_b; normalize(); }
fcn toString{ "%50d / %d".fmt(a,b) }
fcn normalize{ // divide a and b by gcd
g:= a.gcd(b);
a/=g; b/=g;
if(b<0){ a=-a; b=-b; } // denominator > 0
self
}
fcn __opAdd(n){
if(Rational.isChildOf(n)) self(a*n.b + b*n.a, b*n.b); // Rat + Rat
else self(b*n + a, b); // Rat + Int
}
fcn __opSub(n){ self(a*n.b - b*n.a, b*n.b) } // Rat - Rat
fcn __opMul(n){
if(Rational.isChildOf(n)) self(a*n.a, b*n.b); // Rat * Rat
else self(a*n, b); // Rat * Int
}
fcn __opDiv(n){ self(a*n.b,b*n.a) } // Rat / Rat
}

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Task/Bernoulli-numbers/Zkl/bernoulli-numbers-2.zkl Normal file
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var [const] BN=Import.lib("zklBigNum"); // libGMP (GNU MP Bignum Library)
fcn B(N){ // calculate Bernoulli(n)
var A=List.createLong(100,0); // aka static aka not thread safe
foreach m in (N+1){
A[m]=Rational(BN(1),BN(m+1));
foreach j in ([m..1, -1]){ A[j-1]= (A[j-1] - A[j])*j; }
}
A[0]
}

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Task/Bernoulli-numbers/Zkl/bernoulli-numbers-3.zkl Normal file
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foreach b in ([0..1].chain([2..60,2])){ println("B(%2d)%s".fmt(b,B(b))) }