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Ingy döt Net 2023-07-01 11:58:00 -04:00
parent 7387c8f97b
commit cb5bb5e222
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2
Task/Catalan-numbers-Pascals-triangle/00-META.yaml Normal file
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---
from: http://rosettacode.org/wiki/Catalan_numbers/Pascal's_triangle

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Task/Catalan-numbers-Pascals-triangle/00-TASK.txt Normal file
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;Task:
Print out the first   '''15'''   Catalan numbers by extracting them from Pascal's triangle.
;See:
* &nbsp; [https://archive.is/0IrNp Catalan Numbers and the Pascal Triangle]. <!-- Relation Pascal Triangle and the Catalan Numbers Radoslav Jovanovic --> &nbsp; &nbsp; This method enables calculation of Catalan Numbers using only addition and subtraction.
<!-- '''http://milan.milanovic.org/math/english/fibo/fibo4.html is broken. -->
* &nbsp; [http://mathworld.wolfram.com/CatalansTriangle.html Catalan's Triangle] for a Number Triangle that generates Catalan Numbers using only addition.
* &nbsp; Sequence [[oeis:A000108|A000108 on OEIS]] has a lot of information on Catalan Numbers.
;Related Tasks:
[[Pascal's triangle]]
<br><br>

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Task/Catalan-numbers-Pascals-triangle/11l/catalan-numbers-pascals-triangle.11l Normal file
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V n = 15
V t = [0] * (n + 2)
t[1] = 1
L(i) 1 .. n
L(j) (i .< 1).step(-1)
t[j] += t[j - 1]
t[i + 1] = t[i]
L(j) (i + 1 .< 1).step(-1)
t[j] += t[j - 1]
print(t[i + 1] - t[i], end' ‘ ’)

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Task/Catalan-numbers-Pascals-triangle/360-Assembly/catalan-numbers-pascals-triangle.360 Normal file
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CATALAN CSECT
USING CATALAN,R13,R12
SAVEAREA B STM-SAVEAREA(R15)
DC 17F'0'
DC CL8'CATALAN'
STM STM R14,R12,12(R13)
ST R13,4(R15)
ST R15,8(R13)
LR R13,R15
LA R12,4095(R13)
LA R12,1(R12)
* ---- CODE
LA R0,1
ST R0,T t(1)=1
LA R4,0 ix:i=1
LA R6,1 by 1
LH R7,N to n
LOOPI BXH R4,R6,ENDLOOPI loop i
LR R5,R4 ix:j=i+1
LA R5,2(R5) i+2
LA R8,0
BCTR R8,0 by -1
LA R9,1 to 2
LOOP1J BXLE R5,R8,ENLOOP1J loop j
LR R10,R5 j
BCTR R10,0
SLA R10,2
L R2,T(R10) r2=t(j)
LR R1,R10 j
SH R1,=H'4'
L R3,T(R1) r3=t(j-1)
AR R2,R3 r2=r2+r3
ST R2,T(R10) t(j)=t(j)+t(j-1)
B LOOP1J
ENLOOP1J EQU *
LR R1,R4 i
BCTR R1,0
SLA R1,2
L R3,T(R1) t(i)
LA R1,4(R1)
ST R3,T(R1) t(i+1)
LR R5,R4 ix:j=i+2
LA R5,3(R5) i+3
LA R8,0
BCTR R8,0 by -1
LA R9,1 to 2
LOOP2J BXLE R5,R8,ENLOOP2J loop j
LR R10,R5 j
BCTR R10,0
SLA R10,2
L R2,T(R10) r2=t(j)
LR R1,R10 j
SH R1,=H'4'
L R3,T(R1) r3=t(j-1)
AR R2,R3 r2=r2+r3
ST R2,T(R10) t(j)=t(j)+t(j-1)
B LOOP2J
ENLOOP2J EQU *
LR R1,R4 i
BCTR R1,0
SLA R1,2
L R2,T(R1) t(i)
LA R1,4(R1)
L R3,T(R1) t(i+1)
SR R3,R2
CVD R3,P
UNPK Z,P
MVC C,Z
OI C+L'C-1,X'F0'
MVC WTOBUF(8),C+8
WTO MF=(E,WTOMSG)
B LOOPI
ENDLOOPI EQU *
* ---- END CODE
CNOP 0,4
L R13,4(0,R13)
LM R14,R12,12(R13)
XR R15,R15
BR R14
* ---- DATA
N DC H'15'
T DC 17F'0'
P DS PL8
Z DS ZL16
C DS CL16
WTOMSG DS 0F
DC H'80'
DC H'0'
WTOBUF DC CL80' '
YREGS
END

10
Task/Catalan-numbers-Pascals-triangle/ALGOL-68/catalan-numbers-pascals-triangle.alg Normal file
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INT n = 15;
[ 0 : n + 1 ]INT t;
t[0] := 0;
t[1] := 1;
FOR i TO n DO
FOR j FROM i BY -1 TO 2 DO t[j] := t[j] + t[j-1] OD;
t[i+1] := t[i];
FOR j FROM i+1 BY -1 TO 2 DO t[j] := t[j] + t[j-1] OD;
print( ( whole( t[i+1] - t[i], 0 ), " " ) )
OD

20
Task/Catalan-numbers-Pascals-triangle/ALGOL-W/catalan-numbers-pascals-triangle.alg Normal file
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begin
% print the first 15 Catalan numbers from Pascal's triangle %
integer n;
n := 15;
begin
integer array pascalLine ( 1 :: n + 1 );
% the Catalan numbers are the differences between the middle and middle - 1 numbers of the odd %
% lines of Pascal's triangle (lines with 3 or more numbers) %
% note - we only need to calculate the left side of the triangle %
pascalLine( 1 ) := 1;
for c := 2 until n + 1 do begin
% even line %
for i := c - 1 step -1 until 2 do pascalLine( i ) := pascalLine( i - 1 ) + pascalLine( i );
pascalLine( c ) := pascalLine( c - 1 );
% odd line %
for i := c step -1 until 2 do pascalLine( i ) := pascalLine( i - 1 ) + pascalLine( i );
writeon( i_w := 1, s_w := 0, " ", pascalLine( c ) - pascalLine( c - 1 ) )
end for_c
end
end.

2
Task/Catalan-numbers-Pascals-triangle/APL/catalan-numbers-pascals-triangle.apl Normal file
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⍝ Based heavily on the J solution
CATALAN←{¯1↓↑-/1 ¯1↓¨(⊂⎕IO+0 0)⍉¨0 2⌽¨⊂(⎕IO-⍨⍳N){+\⍣⍺⊢⍵}⍤0 1⊢1⍴⍨N←⍵+2}

16
Task/Catalan-numbers-Pascals-triangle/AWK/catalan-numbers-pascals-triangle.awk Normal file
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# syntax: GAWK -f CATALAN_NUMBERS_PASCALS_TRIANGLE.AWK
# converted from C
BEGIN {
printf("1")
for (n=2; n<=15; n++) {
num = den = 1
for (k=2; k<=n; k++) {
num *= (n + k)
den *= k
catalan = num / den
}
printf(" %d",catalan)
}
printf("\n")
exit(0)
}

48
Task/Catalan-numbers-Pascals-triangle/Action-/catalan-numbers-pascals-triangle.action Normal file
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INCLUDE "D2:REAL.ACT" ;from the Action! Tool Ki
DEFINE PTR="CARD"
DEFINE REALSIZE="6"
PTR FUNC GetItemAddr(PTR buf BYTE i)
RETURN (buf+REALSIZE*i)
PROC Main()
DEFINE COUNT="15"
BYTE ARRAY buf(102) ;(COUNT+2)*REALSIZE
REAL POINTER r1,r2
REAL c
BYTE i,j
Put(125) PutE() ;clear the screen
r1=GetItemAddr(buf,1)
IntToReal(1,r1)
FOR i=1 TO COUNT
DO
j=i+1
WHILE j>=2
DO
r1=GetItemAddr(buf,j)
r2=GetItemAddr(buf,j-1)
RealAdd(r1,r2,r1) ;t(j)==+t(j-1)
j==-1
OD
r1=GetItemAddr(buf,i)
r2=GetItemAddr(buf,i+1)
RealAssign(r1,r2) ;t(i+1)=t(i)
j=i+2
WHILE j>=2
DO
r1=GetItemAddr(buf,j)
r2=GetItemAddr(buf,j-1)
RealAdd(r1,r2,r1) ;t(j)==+t(j-1)
j==-1
OD
r1=GetItemAddr(buf,i)
r2=GetItemAddr(buf,i+1)
RealSub(r2,r1,c) ;c=t(i+1)-t(i)
PrintF("C(%B)=",i) PrintRE(c)
OD
RETURN

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Task/Catalan-numbers-Pascals-triangle/Ada/catalan-numbers-pascals-triangle.ada Normal file
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with Ada.Text_IO, Pascal;
procedure Catalan is
Last: Positive := 15;
Row: Pascal.Row := Pascal.First_Row(2*Last+1);
begin
for I in 1 .. Last loop
Row := Pascal.Next_Row(Row);
Row := Pascal.Next_Row(Row);
Ada.Text_IO.Put(Integer'Image(Row(I+1)-Row(I+2)));
end loop;
end Catalan;

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Task/Catalan-numbers-Pascals-triangle/Arturo/catalan-numbers-pascals-triangle.arturo Normal file
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n: 15
t: new array.of:n+2 0
t\[1]: 1
loop 1..n 'i [
loop i..1 'j -> t\[j]: t\[j] + t\[j-1]
t\[i+1]: t\[i]
loop (i+1)..1 'j -> t\[j]: t\[j] + t\[j-1]
prints t\[i+1] - t\[i]
prints " "
]
print ""

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Task/Catalan-numbers-Pascals-triangle/AutoHotkey/catalan-numbers-pascals-triangle.ahk Normal file
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/* Generate Catalan Numbers
//
// smgs: 20th Feb, 2014
*/
Array := [], Array[2,1] := Array[2,2] := 1 ; Array inititated and 2nd row of pascal's triangle assigned
INI := 3 ; starts with calculating the 3rd row and as such the value
Loop, 31 ; every odd row is taken for calculating catalan number as such to obtain 15 we need 2n+1
{
if ( A_index > 2 )
{
Loop, % A_INDEX
{
old := ini-1, index := A_index, index_1 := A_index + 1
Array[ini, index_1] := Array[old, index] + Array[old, index_1]
Array[ini, 1] := Array[ini, ini] := 1
line .= Array[ini, A_index] " "
}
;~ MsgBox % line ; gives rows of pascal's triangle
; calculating every odd row starting from 1st so as to obtain catalan's numbers
if ( mod(ini,2) != 0)
{
StringSplit, res, line, %A_Space%
ans := res0//2, ans_1 := ans++
result := result . res%ans_1% - res%ans% " "
}
line :=
ini++
}
}
MsgBox % result

22
Task/Catalan-numbers-Pascals-triangle/Batch-File/catalan-numbers-pascals-triangle.bat Normal file
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@echo off
setlocal ENABLEDELAYEDEXPANSION
set n=15
set /A nn=n+1
for /L %%i in (0,1,%nn%) do set t.%%i=0
set t.1=1
for /L %%i in (1,1,%n%) do (
set /A ip=%%i+1
for /L %%j in (%%i,-1,1) do (
set /A jm=%%j-1
set /A t.%%j=t.%%j+t.!jm!
)
set /A t.!ip!=t.%%i
for /L %%j in (!ip!,-1,1) do (
set /A jm=%%j-1
set /A t.%%j=t.%%j+t.!jm!
)
set /A ci=t.!ip!-t.%%i
echo !ci!
)
)
pause

16
Task/Catalan-numbers-Pascals-triangle/C++/catalan-numbers-pascals-triangle.cpp Normal file
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// Generate Catalan Numbers
//
// Nigel Galloway: June 9th., 2012
//
#include <iostream>
int main() {
const int N = 15;
int t[N+2] = {0,1};
for(int i = 1; i<=N; i++){
for(int j = i; j>1; j--) t[j] = t[j] + t[j-1];
t[i+1] = t[i];
for(int j = i+1; j>1; j--) t[j] = t[j] + t[j-1];
std::cout << t[i+1] - t[i] << " ";
}
return 0;
}

9
Task/Catalan-numbers-Pascals-triangle/C-sharp/catalan-numbers-pascals-triangle.cs Normal file
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int n = 15;
List<int> t = new List<int>() { 0, 1 };
for (int i = 1; i <= n; i++)
{
for (var j = i; j > 1; j--) t[j] += t[j - 1];
t.Add(t[i]);
for (var j = i + 1; j > 1; j--) t[j] += t[j - 1];
Console.Write(((i == 1) ? "" : ", ") + (t[i + 1] - t[i]));
}

45
Task/Catalan-numbers-Pascals-triangle/C/catalan-numbers-pascals-triangle.c Normal file
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//This code implements the print of 15 first Catalan's Numbers
//Formula used:
// __n__
// | | (n + k) / k n>0
// k=2
#include <stdio.h>
#include <stdlib.h>
//the number of Catalan's Numbers to be printed
const int N = 15;
int main()
{
//loop variables (in registers)
register int k, n;
//necessarily ull for reach big values
unsigned long long int num, den;
//the nmmber
int catalan;
//the first is not calculated for the formula
printf("1 ");
//iterating from 2 to 15
for (n=2; n<=N; ++n) {
//initializaing for products
num = den = 1;
//applying the formula
for (k=2; k<=n; ++k) {
num *= (n+k);
den *= k;
catalan = num /den;
}
//output
printf("%d ", catalan);
}
//the end
printf("\n");
return 0;
}

10
Task/Catalan-numbers-Pascals-triangle/Common-Lisp/catalan-numbers-pascals-triangle.lisp Normal file
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(defun catalan (n)
"Return the n-th Catalan number"
(if (<= n 1) 1
(let ((result 2))
(dotimes (k (- n 2) result)
(setq result (* result (/ (+ n k 2) (+ k 2)))) ))))
(dotimes (n 15)
(print (catalan (1+ n))) )

16
Task/Catalan-numbers-Pascals-triangle/D/catalan-numbers-pascals-triangle.d Normal file
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void main() {
import std.stdio;
enum uint N = 15;
uint[N + 2] t;
t[1] = 1;
foreach (immutable i; 1 .. N + 1) {
foreach_reverse (immutable j; 2 .. i + 1)
t[j] += t[j - 1];
t[i + 1] = t[i];
foreach_reverse (immutable j; 2 .. i + 2)
t[j] += t[j - 1];
write(t[i + 1] - t[i], ' ');
}
}

59
Task/Catalan-numbers-Pascals-triangle/EDSAC-order-code/catalan-numbers-pascals-triangle.edsac Normal file
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[Catalan numbers from Pascal triangle, Rosetta Code website.
EDSAC program, Initial Orders 2]
..PZ [blank tape and terminator]
T54K [refer to working array with 'C']
P300F [address of working array]
T46K [to call print subroutine with 'G N']
P56F [address of print subroutine]
[Modification of library subroutine P7.
Prints non-negative integer, up to 10 digits, right-justified.
55 locations, load at even address.]
E25KTN
GKA3FT42@A47@T31@ADE10@T31@A48@T31@SDTDH44#@NDYFLDT4DS43@
TFH17@S17@A43@G23@UFS43@T1FV4DAFG50@SFLDUFXFOFFFSFL4FT4DA49@
T31@A1FA43@G20@XFP1024FP610D@524D!FO46@O26@XFO46@SFL8FT4DE39@
[Main program]
PK T200K GK
[Constants]
[0] PD [short constant 1]
[1] P2F [to inc address by 2]
[2] T#C [used in manufacturing EDSAC orders]
[3] MF [add to T order to make A order with same address]
[4] #F [set figures]
[5] &F [line feed]
[6] @F [carriage return]
[7] P7D [maximum n = 15]
[Variable]
[8] PF [n]
[Enter with acc = 0]
[9] O4@ [set teleprinter to figures]
T4#C T2#C T#C A@ TC [initialize first 3 terms to 1, 0, 0]
T8@ E58@ [set n := 0; jump to inc n and print C_n]
[Outer loop; here with n updated]
[17] TF A8@ [acc := latest n]
L1F A2@ T22@ [make and store order 'T 2n #C']
[22] T#C [sets term := 0; also used to test for end of loop]
A2@ [load 'T#C', initial value of order 31]
[Loop to convert e.g. (20, 15, 6, 1) to (35, 21, 7, 1); works left to right]
[24] U31@ A3@ U29@ A1@ T30@ [set up orders on next line]
[29] A#C A#C T#C [replaced by manufactured orders]
A31@ A1@ S22@ E38@ [inc address in order 31, jump out if done]
A22@ E24@ [not done, loop back]
[38] A22@ T48@ [initialize order 48]
[Loop to convert e.g. (35, 21, 7, 1) to (70, 56, 28, 8, 1); works right to left]
[40] TF A48@ A3@ U46@ S1@ T47@ [set up orders on next line]
[46] A#C A#C T#C [replaced by manufactured orders]
A48@ S1@ T48@ [dec address in order 48]
A2@ S48@ G40@ [test for done, loop back if not]
A#C LD T#C [double first term, e.g. 35 -> 70 (not done in loop)]
[Increment n and print Catalan number C_n]
[58] TD [clear 0D, ensures sandwich bit = 0]
A8@ A@ U8@ TF [inc n; set 0D := n by setting 0F := n]
A63@ GN [print n]
A#C S4#C TD A68@ GN [print Catalan number C_n, e.g. C_5 = 70 - 28 = 42]
O6@ O5@ [print CR, LF]
A8@ S7@ G17@ [test for maximum n, loop back if not]
[75] O4@ ZF [flush printer buffer; stop]
E9Z PF [define entry point; enter with acc = 0]

38
Task/Catalan-numbers-Pascals-triangle/ERRE/catalan-numbers-pascals-triangle.erre Normal file
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PROGRAM CATALAN
!$DOUBLE
DIM CATALAN[50]
FUNCTION ODD(X)
ODD=FRC(X/2)<>0
END FUNCTION
PROCEDURE GETCATALAN(L)
LOCAL J,K,W
LOCAL DIM PASTRI[100]
L=L*2
PASTRI[0]=1
J=0
WHILE J<L DO
J+=1
K=INT((J+1)/2)
PASTRI[K]=PASTRI[K-1]
FOR W=K TO 1 STEP -1 DO
PASTRI[W]+=PASTRI[W-1]
END FOR
IF NOT(ODD(J)) THEN
K=INT(J/2)
CATALAN[K]=PASTRI[K]-PASTRI[K-1]
END IF
END WHILE
END PROCEDURE
BEGIN
LL=15
GETCATALAN(LL)
FOR I=1 TO LL DO
WRITE("### ####################";I;CATALAN[I])
END FOR
END PROGRAM

15
Task/Catalan-numbers-Pascals-triangle/EchoLisp/catalan-numbers-pascals-triangle-1.l Normal file
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(define dim 100)
(define-syntax-rule (Tidx i j) (+ i (* dim j)))
;; generates Catalan's triangle
;; T (i , j) = T(i-1,j) + T (i, j-1)
(define (T n)
(define i (modulo n dim))
(define j (quotient n dim))
(cond
((zero? i) 1) ;; left column = 1
((= i j) (T (Tidx (1- i) j))) ;; diagonal value = left value
(else (+ (T (Tidx (1- i) j)) (T (Tidx i (1- j)))))))
(remember 'T #(1))

4
Task/Catalan-numbers-Pascals-triangle/EchoLisp/catalan-numbers-pascals-triangle-2.l Normal file
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;; take elements on diagonal = Catalan numbers
(for ((i (in-range 0 16))) (write (T (Tidx i i))))
→ 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

15
Task/Catalan-numbers-Pascals-triangle/Elixir/catalan-numbers-pascals-triangle.elixir Normal file
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defmodule Catalan do
def numbers(num) do
{result,_} = Enum.reduce(1..num, {[],{0,1}}, fn i,{list,t0} ->
t1 = numbers(i, t0)
t2 = numbers(i+1, Tuple.insert_at(t1, i+1, elem(t1, i)))
{[elem(t2, i+1) - elem(t2, i) | list], t2}
end)
Enum.reverse(result)
end
defp numbers(0, t), do: t
defp numbers(n, t), do: numbers(n-1, put_elem(t, n, elem(t, n-1) + elem(t, n)))
end
IO.inspect Catalan.numbers(15)

17
Task/Catalan-numbers-Pascals-triangle/Erlang/catalan-numbers-pascals-triangle.erl Normal file
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-module(catalin).
-compile(export_all).
mul(N,D,S,S)->
N2=N*(S+S),
D2=D*S,
K = N2 div D2 ;
mul(N,D,S,L)->
N2=N*(S+L),
D2=D*L,
K = mul(N2,D2,S,L+1).
catl(Ans,16) -> Ans;
catl(D,S)->
C=mul(1,1,S,2),
catl([D|C],S+1).
main()->
Ans=catl(1,2).

15
Task/Catalan-numbers-Pascals-triangle/F-Sharp/catalan-numbers-pascals-triangle.fs Normal file
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let mutable nm=uint64(1)
let mutable dm=uint64(1)
let mutable a=uint64(1)
printf "1, "
for i = 2 to 15 do
nm<-uint64(1)
dm<-uint64(1)
for k = 2 to i do
nm <-uint64( uint64(nm) * (uint64(i)+uint64(k)))
dm <-uint64( uint64(dm) * uint64(k))
let a = uint64(uint64(nm)/uint64(dm))
printf "%u"a
if(i<>15) then
printf ", "

14
Task/Catalan-numbers-Pascals-triangle/Factor/catalan-numbers-pascals-triangle.factor Normal file
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USING: arrays grouping io kernel math prettyprint sequences ;
IN: rosetta-code.catalan-pascal
: next-row ( seq -- seq' )
2 clump [ sum ] map 1 prefix 1 suffix ;
: pascal ( n -- seq )
1 - { { 1 } } swap [ dup last next-row suffix ] times ;
15 2 * pascal [ length odd? ] filter [
dup length 1 = [ 1 ]
[ dup midpoint@ dup 1 + 2array swap nths first2 - ] if
pprint bl
] each drop

56
Task/Catalan-numbers-Pascals-triangle/FreeBASIC/catalan-numbers-pascals-triangle.basic Normal file
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' version 15-09-2015
' compile with: fbc -s console
#Define size 31 ' (N * 2 + 1)
Sub pascal_triangle(rows As Integer, Pas_tri() As ULongInt)
Dim As Integer x, y
For x = 1 To rows
Pas_tri(1,x) = 1
Pas_tri(x,1) = 1
Next
For x = 2 To rows
For y = 2 To rows + 1 - x
Pas_tri(x, y) = pas_tri(x - 1 , y) + pas_tri(x, y - 1)
Next
Next
End Sub
' ------=< MAIN >=------
Dim As Integer count, row
Dim As ULongInt triangle(1 To size, 1 To size)
pascal_triangle(size, triangle())
' 1 1 1 1 1 1
' 1 2 3 4 5 6
' 1 3 6 10 15 21
' 1 4 10 20 35 56
' 1 5 15 35 70 126
' 1 6 21 56 126 252
' The Pascal triangle is rotated 45 deg.
' to find the Catalan number we need to follow the diagonal
' for top left to bottom right
' take the number on diagonal and subtract the number in de cell
' one up and one to right
' 1 (2 - 1), 2 (6 - 4), 5 (20 - 15) ...
Print "The first 15 Catalan numbers are" : print
count = 1 : row = 2
Do
Print Using "###: #########"; count; triangle(row, row) - triangle(row +1, row -1)
row = row + 1
count = count + 1
Loop Until count > 15
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End

18
Task/Catalan-numbers-Pascals-triangle/Go/catalan-numbers-pascals-triangle.go Normal file
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package main
import "fmt"
func main() {
const n = 15
t := [n + 2]uint64{0, 1}
for i := 1; i <= n; i++ {
for j := i; j > 1; j-- {
t[j] += t[j-1]
}
t[i+1] = t[i]
for j := i + 1; j > 1; j-- {
t[j] += t[j-1]
}
fmt.Printf("%2d : %d\n", i, t[i+1]-t[i])
}
}

24
Task/Catalan-numbers-Pascals-triangle/Groovy/catalan-numbers-pascals-triangle.groovy Normal file
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class Catalan
{
public static void main(String[] args)
{
BigInteger N = 15;
BigInteger k,n,num,den;
BigInteger catalan;
print(1);
for(n=2;n<=N;n++)
{
num = 1;
den = 1;
for(k=2;k<=n;k++)
{
num = num*(n+k);
den = den*k;
catalan = num/den;
}
print(" " + catalan);
}
}
}
​

19
Task/Catalan-numbers-Pascals-triangle/Haskell/catalan-numbers-pascals-triangle.hs Normal file
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import System.Environment (getArgs)
-- Pascal's triangle.
pascal :: [[Integer]]
pascal = [1] : map (\row -> 1 : zipWith (+) row (tail row) ++ [1]) pascal
-- The Catalan numbers from Pascal's triangle. This uses a method from
-- http://www.cut-the-knot.org/arithmetic/algebra/CatalanInPascal.shtml
-- (see "Grimaldi").
catalan :: [Integer]
catalan = map (diff . uncurry drop) $ zip [0..] (alt pascal)
where alt (x:_:zs) = x : alt zs -- every other element of an infinite list
diff (x:y:_) = x - y
diff (x:_) = x
main :: IO ()
main = do
ns <- fmap (map read) getArgs :: IO [Int]
mapM_ (print . flip take catalan) ns

6
Task/Catalan-numbers-Pascals-triangle/Icon/catalan-numbers-pascals-triangle.icon Normal file
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link math
procedure main(A)
limit := (integer(A[1])|15)+1
every write(right(binocoef(i := 2*seq(0)\limit,i/2)-binocoef(i,i/2+1),30))
end

1
Task/Catalan-numbers-Pascals-triangle/J/catalan-numbers-pascals-triangle-1.j Normal file
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Catalan=. }:@:(}.@:((<0 1)&|:) - }:@:((<0 1)&|:@:(2&|.)))@:(i. +/\@]^:[ #&1)@:(2&+)

2
Task/Catalan-numbers-Pascals-triangle/J/catalan-numbers-pascals-triangle-2.j Normal file
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Catalan 15
1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

17
Task/Catalan-numbers-Pascals-triangle/J/catalan-numbers-pascals-triangle-3.j Normal file
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o=. @: NB. Composition of verbs (functions)
( PascalTriangle=. i. ((+/\@]^:[)) #&1 ) 5
1 1 1 1 1
1 2 3 4 5
1 3 6 10 15
1 4 10 20 35
1 5 15 35 70
( MiddleDiagonal=. (<0 1)&|: ) o PascalTriangle 5
1 2 6 20 70
( AdjacentLeft=. MiddleDiagonal o (2&|.) ) o PascalTriangle 5
1 4 15 1 5
( Catalan=. }: o (}. o MiddleDiagonal - }: o AdjacentLeft) o PascalTriangle o (2&+) f. ) 5
1 2 5 14 42
Catalan
}:@:(}.@:((<0 1)&|:) - }:@:((<0 1)&|:@:(2&|.)))@:(i. +/\@]^:[ #&1)@:(2&+)

20
Task/Catalan-numbers-Pascals-triangle/Java/catalan-numbers-pascals-triangle.java Normal file
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public class Test {
public static void main(String[] args) {
int N = 15;
int[] t = new int[N + 2];
t[1] = 1;
for (int i = 1; i <= N; i++) {
for (int j = i; j > 1; j--)
t[j] = t[j] + t[j - 1];
t[i + 1] = t[i];
for (int j = i + 1; j > 1; j--)
t[j] = t[j] + t[j - 1];
System.out.printf("%d ", t[i + 1] - t[i]);
}
}
}

7
Task/Catalan-numbers-Pascals-triangle/JavaScript/catalan-numbers-pascals-triangle-1.js Normal file
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var n = 15;
for (var t = [0, 1], i = 1; i <= n; i++) {
for (var j = i; j > 1; j--) t[j] += t[j - 1];
t[i + 1] = t[i];
for (var j = i + 1; j > 1; j--) t[j] += t[j - 1];
document.write(i == 1 ? '' : ', ', t[i + 1] - t[i]);
}

64
Task/Catalan-numbers-Pascals-triangle/JavaScript/catalan-numbers-pascals-triangle-2.js Normal file
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(() => {
'use strict';
// CATALAN
// catalanSeries :: Int -> [Int]
let catalanSeries = n => {
let alternate = xs => xs.reduce(
(a, x, i) => i % 2 === 0 ? a.concat([x]) : a, []
),
diff = xs => xs.length > 1 ? xs[0] - xs[1] : xs[0];
return alternate(pascal(n * 2))
.map((xs, i) => diff(drop(i, xs)));
}
// PASCAL
// pascal :: Int -> [[Int]]
let pascal = n => until(
m => m.level <= 1,
m => {
let nxt = zipWith(
(a, b) => a + b, [0].concat(m.row), m.row.concat(0)
);
return {
row: nxt,
triangle: m.triangle.concat([nxt]),
level: m.level - 1
}
}, {
level: n,
row: [1],
triangle: [
[1]
]
}
)
.triangle;
// GENERIC FUNCTIONS
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
let zipWith = (f, xs, ys) =>
xs.length === ys.length ? (
xs.map((x, i) => f(x, ys[i]))
) : undefined;
// until :: (a -> Bool) -> (a -> a) -> a -> a
let until = (p, f, x) => {
let v = x;
while (!p(v)) v = f(v);
return v;
}
// drop :: Int -> [a] -> [a]
let drop = (n, xs) => xs.slice(n);
// tail :: [a] -> [a]
let tail = xs => xs.length ? xs.slice(1) : undefined;
return tail(catalanSeries(16));
})();

1
Task/Catalan-numbers-Pascals-triangle/JavaScript/catalan-numbers-pascals-triangle-3.js Normal file
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[1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,9694845]

7
Task/Catalan-numbers-Pascals-triangle/Jq/catalan-numbers-pascals-triangle-1.jq Normal file
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def binomial(n; k):
if k > n / 2 then binomial(n; n-k)
else reduce range(1; k+1) as $i (1; . * (n - $i + 1) / $i)
end;
# Direct (naive) computation using two numbers in Pascal's triangle:
def catalan_by_pascal: . as $n | binomial(2*$n; $n) - binomial(2*$n; $n-1);

21
Task/Catalan-numbers-Pascals-triangle/Jq/catalan-numbers-pascals-triangle-2.jq Normal file
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$ jq -n -c -f Catalan_numbers_Pascal.jq
[0,0]
[1,1]
[2,2]
[3,5]
[4,14]
[5,42]
[6,132]
[7,429]
[8,1430]
[9,4862]
[10,16796]
[11,58786]
[12,208012]
[13,742900]
[14,2674440]
[15,9694845]
[30,3814986502092304]
[31,14544636039226880]
[510,5.491717746183512e+302]
[511,null]

16
Task/Catalan-numbers-Pascals-triangle/Julia/catalan-numbers-pascals-triangle.julia Normal file
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# v0.6
function pascal(n::Int)
r = ones(Int, n, n)
for i in 2:n, j in 2:n
r[i, j] = r[i-1, j] + r[i, j-1]
end
return r
end
function catalan_num(n::Int)
p = pascal(n + 2)
p[n+4:n+3:end-1] - diag(p, 2)
end
@show catalan_num(15)

25
Task/Catalan-numbers-Pascals-triangle/Kotlin/catalan-numbers-pascals-triangle.kotlin Normal file
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// version 1.1.2
import java.math.BigInteger
val ONE = BigInteger.ONE
fun pascal(n: Int, k: Int): BigInteger {
if (n == 0 || k == 0) return ONE
val num = (k + 1..n).fold(ONE) { acc, i -> acc * BigInteger.valueOf(i.toLong()) }
val den = (2..n - k).fold(ONE) { acc, i -> acc * BigInteger.valueOf(i.toLong()) }
return num / den
}
fun catalanFromPascal(n: Int) {
for (i in 1 until n step 2) {
val mi = i / 2 + 1
val catalan = pascal(i, mi) - pascal(i, mi - 2)
println("${"%2d".format(mi)} : $catalan")
}
}
fun main(args: Array<String>) {
val n = 15
catalanFromPascal(n * 2)
}

17
Task/Catalan-numbers-Pascals-triangle/Lua/catalan-numbers-pascals-triangle.lua Normal file
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function nextrow (t)
local ret = {}
t[0], t[#t + 1] = 0, 0
for i = 1, #t do ret[i] = t[i - 1] + t[i] end
return ret
end
function catalans (n)
local t, middle = {1}
for i = 1, n do
middle = math.ceil(#t / 2)
io.write(t[middle] - (t[middle + 1] or 0) .. " ")
t = nextrow(nextrow(t))
end
end
catalans(15)

33
Task/Catalan-numbers-Pascals-triangle/M2000-Interpreter/catalan-numbers-pascals-triangle.m2000 Normal file
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Module CatalanNumbers {
Def Integer count, t_row, size=31
Dim triangle(1 to size, 1 to size)
\\ call sub
pascal_triangle(size, &triangle())
Print "The first 15 Catalan numbers are"
count = 1% : t_row = 2%
Do {
Print Format$("{0:0:-3}:{1:0:-15}", count, triangle(t_row, t_row) - triangle(t_row +1, t_row -1))
t_row++
count++
} Until count > 15
End
Sub pascal_triangle(rows As Integer, &Pas_tri())
Local x=0%, y=0%
For x = 1 To rows
Pas_tri( 1%, x ) = 1@
Pas_tri( x, 1% ) = 1@
Next x
if rows<2 then exit sub
For x = 2 To rows-1
For y = 2 To rows + 1 - x
Pas_tri(x, y) = pas_tri(x - 1 , y) + pas_tri(x, y - 1)
Next y
Next x
End Sub
}
CatalanNumbers

3
Task/Catalan-numbers-Pascals-triangle/MATLAB/catalan-numbers-pascals-triangle.m Normal file
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n = 15;
p = pascal(n + 2);
p(n + 4 : n + 3 : end - 1)' - diag(p, 2)

12
Task/Catalan-numbers-Pascals-triangle/Maple/catalan-numbers-pascals-triangle.maple Normal file
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catalan:=proc(n)
local i,a:=[1],C:=[1];
for i to n do
a:=[0,op(a)]+[op(a),0];
a:=[0,op(a)]+[op(a),0];
C:=[op(C),a[i+1]-a[i]];
od;
C
end:
catalan(10);
# [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]

16
Task/Catalan-numbers-Pascals-triangle/Mathematica/catalan-numbers-pascals-triangle.math Normal file
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nextrow[lastrow_] := Module[{output},
output = ConstantArray[1, Length[lastrow] + 1];
Do[
output[[i + 1]] = lastrow[[i]] + lastrow[[i + 1]];
, {i, 1, Length[lastrow] - 1}];
output
]
pascaltriangle[size_] := NestList[nextrow, {1}, size]
catalannumbers[length_] := Module[{output, basetriangle},
basetriangle = pascaltriangle[2 length];
list1 = basetriangle[[# *2 + 1, # + 1]] & /@ Range[length];
list2 = basetriangle[[# *2 + 1, # + 2]] & /@ Range[length];
list1 - list2
]
(* testing *)
catalannumbers[15]

10
Task/Catalan-numbers-Pascals-triangle/Nim/catalan-numbers-pascals-triangle.nim Normal file
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const n = 15
var t = newSeq[int](n + 2)
t[1] = 1
for i in 1..n:
for j in countdown(i, 1): t[j] += t[j-1]
t[i+1] = t[i]
for j in countdown(i+1, 1): t[j] += t[j-1]
stdout.write t[i+1] - t[i], " "
stdout.write '\n'

12
Task/Catalan-numbers-Pascals-triangle/OCaml/catalan-numbers-pascals-triangle.ocaml Normal file
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let catalan : int ref = ref 0 in
Printf.printf "%d ," 1 ;
for i = 2 to 9 do
let nm : int ref = ref 1 in
let den : int ref = ref 1 in
for k = 2 to i do
nm := (!nm)*(i+k);
den := (!den)*k;
catalan := (!nm)/(!den) ;
done;
print_int (!catalan); print_string "," ;
done;;

7
Task/Catalan-numbers-Pascals-triangle/Oforth/catalan-numbers-pascals-triangle.fth Normal file
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import: mapping
: pascal( n -- [] )
[ 1 ] n #[ dup [ 0 ] + [ 0 ] rot + zipWith( #+ ) ] times ;
: catalan( n -- m )
n 2 * pascal at( n 1+ ) n 1+ / ;

1
Task/Catalan-numbers-Pascals-triangle/PARI-GP/catalan-numbers-pascals-triangle.parigp Normal file
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vector(15,n,binomial(2*n,n)-binomial(2*n,n+1))

36
Task/Catalan-numbers-Pascals-triangle/Pascal/catalan-numbers-pascals-triangle.pas Normal file
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Program CatalanNumbers
type
tElement = Uint64;
var
Catalan : array[0..50] of tElement;
procedure GetCatalan(L:longint);
var
PasTri : array[0..100] of tElement;
j,k: longInt;
begin
l := l*2;
PasTri[0] := 1;
j := 0;
while (j<L) do
begin
inc(j);
k := (j+1) div 2;
PasTri[k] :=PasTri[k-1];
For k := k downto 1 do
inc(PasTri[k],PasTri[k-1]);
IF NOT(Odd(j)) then
begin
k := j div 2;
Catalan[k] :=PasTri[k]-PasTri[k-1];
end;
end;
end;
var
i,l: longint;
Begin
l := 15;
GetCatalan(L);
For i := 1 to L do
Writeln(i:3,Catalan[i]:20);
end.

8
Task/Catalan-numbers-Pascals-triangle/Perl/catalan-numbers-pascals-triangle-1.pl Normal file
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use constant N => 15;
my @t = (0, 1);
for(my $i = 1; $i <= N; $i++) {
for(my $j = $i; $j > 1; $j--) { $t[$j] += $t[$j-1] }
$t[$i+1] = $t[$i];
for(my $j = $i+1; $j>1; $j--) { $t[$j] += $t[$j-1] }
print $t[$i+1] - $t[$i], " ";
}

2
Task/Catalan-numbers-Pascals-triangle/Perl/catalan-numbers-pascals-triangle-2.pl Normal file
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use ntheory qw/binomial/;
print join(" ", map { binomial( 2*$_, $_) / ($_+1) } 1 .. 1000), "\n";

16
Task/Catalan-numbers-Pascals-triangle/Phix/catalan-numbers-pascals-triangle-1.phix Normal file
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(phixonline)-->
<span style="color: #008080;">constant</span> <span style="color: #000000;">N</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">15</span> <span style="color: #000080;font-style:italic;">-- accurate to 30, nan/inf for anything over 514 (gmp version is below).</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">catalan</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{},</span> <span style="color: #000080;font-style:italic;">-- (&gt;=1 only)</span>
<span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</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: #004080;">atom</span> <span style="color: #000000;">p1</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</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: #008080;">do</span>
<span style="color: #000000;">p1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">2</span>
<span style="color: #000000;">catalan</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">catalan</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</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;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">N</span><span style="color: #0000FF;">-</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">p1</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">p</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;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000080;font-style:italic;">-- ?p[1..N-i+1]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #0000FF;">?</span><span style="color: #000000;">catalan</span>
<!--

33
Task/Catalan-numbers-Pascals-triangle/Phix/catalan-numbers-pascals-triangle-2.phix Normal file
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-- FreeBASIC said:
--' 1 1 1 1 1 1
--' 1 2 3 4 5 6
--' 1 3 6 10 15 21
--' 1 4 10 20 35 56
--' 1 5 15 35 70 126
--' 1 6 21 56 126 252
--' The Pascal triangle is rotated 45 deg.
--' to find the Catalan number we need to follow the diagonal
--' for top left to bottom right
--' take the number on diagonal and subtract the number in de cell
--' one up and one to right
--' 1 (2 - 1), 2 (6 - 4), 5 (20 - 15) ...
--
-- The first thing that struck me was it is twice as big as it needs to be,
-- something like this would do...
-- 1 1 1 1 1 1
-- 2 3 4 5 6
-- 6 10 15 21
-- 20 35 56
-- 70 126
-- 252
-- It is more obvious from the upper square that the diagonal on that, which is
-- that same as column 1 on this, is twice the previous, which on the second
-- diagram is in column 2. Further, once we have calculated the value for column
-- one above, we can use it immediately to calculate the next catalan number and
-- do not need to store it. Lastly we can overwrite row 1 with row 2 etc in situ,
-- and the following shows what we need for subsequent rounds:
-- 1 1 1 1 1
-- 3 4 5 6
-- 10 15 21
-- 35 56
-- 126 (unused)

23
Task/Catalan-numbers-Pascals-triangle/Phix/catalan-numbers-pascals-triangle-3.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;">function</span> <span style="color: #000000;">catalanB</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: #000080;font-style:italic;">-- very very fast!</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">catalan</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_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: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">p1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">p1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</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: #008080;">do</span>
<span style="color: #7060A8;">mpz_mul_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">mpz_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">catalan</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</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;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</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;">mpz_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span><span style="color: #000000;">p1</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: #008080;">return</span> <span style="color: #000000;">catalan</span><span style="color: #0000FF;">[</span><span style="color: #000000;">n</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</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;">"%d: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">100</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">shorten</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">catalanB</span><span style="color: #0000FF;">(</span><span style="color: #000000;">100</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;">"%d: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">250</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">shorten</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">catalanB</span><span style="color: #0000FF;">(</span><span style="color: #000000;">250</span><span style="color: #0000FF;">)))})</span>
<!--

14
Task/Catalan-numbers-Pascals-triangle/PicoLisp/catalan-numbers-pascals-triangle.l Normal file
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(de bino (N K)
(let f
'((N)
(if (=0 N) 1 (apply * (range 1 N))) )
(/
(f N)
(* (f (- N K)) (f K)) ) ) )
(for N 15
(println
(-
(bino (* 2 N) N)
(bino (* 2 N) (inc N)) ) ) )
(bye)

26
Task/Catalan-numbers-Pascals-triangle/PureBasic/catalan-numbers-pascals-triangle.basic Normal file
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#MAXNUM = 15
Declare catalan()
If OpenConsole("Catalan numbers")
catalan()
Input()
End 0
Else
End -1
EndIf
Procedure catalan()
Define k.i, n.i, num.d, den.d, cat.d
Print("1 ")
For n=2 To #MAXNUM
num=1 : den =1
For k=2 To n
num * (n+k)
den * k
cat = num / den
Next
Print(Str(cat)+" ")
Next
EndProcedure

12
Task/Catalan-numbers-Pascals-triangle/Python/catalan-numbers-pascals-triangle-1.py Normal file
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>>> n = 15
>>> t = [0] * (n + 2)
>>> t[1] = 1
>>> for i in range(1, n + 1):
for j in range(i, 1, -1): t[j] += t[j - 1]
t[i + 1] = t[i]
for j in range(i + 1, 1, -1): t[j] += t[j - 1]
print(t[i+1] - t[i], end=' ')
1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845
>>>

9
Task/Catalan-numbers-Pascals-triangle/Python/catalan-numbers-pascals-triangle-2.py Normal file
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def catalan_number(n):
nm = dm = 1
for k in range(2, n+1):
nm, dm = ( nm*(n+k), dm*k )
return nm/dm
print [catalan_number(n) for n in range(1, 16)]
[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845]

134
Task/Catalan-numbers-Pascals-triangle/Python/catalan-numbers-pascals-triangle-3.py Normal file
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'''Catalan numbers from Pascal's triangle'''
from itertools import (islice)
from operator import (add)
# nCatalans :: Int -> [Int]
def nCatalans(n):
'''The first n Catalan numbers,
derived from Pascal's triangle.'''
# diff :: [Int] -> Int
def diff(xs):
'''Difference between the first two items in the list,
if its length is more than one.
Otherwise, the first (only) item in the list.'''
return (
xs[0] - (xs[1] if 1 < len(xs) else 0)
) if xs else None
return list(map(
compose(diff)(uncurry(drop)),
enumerate(map(fst, take(n)(
everyOther(
pascalTriangle()
)
)))
))
# pascalTriangle :: Gen [[Int]]
def pascalTriangle():
'''A non-finite stream of
Pascal's triangle rows.'''
return iterate(nextPascal)([1])
# nextPascal :: [Int] -> [Int]
def nextPascal(xs):
'''A row of Pascal's triangle
derived from a preceding row.'''
return zipWith(add)([0] + xs)(xs + [0])
# TEST ----------------------------------------------------
# main :: IO ()
def main():
'''First 16 Catalan numbers.'''
print(
nCatalans(16)
)
# GENERIC -------------------------------------------------
# compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
def compose(g):
'''Right to left function composition.'''
return lambda f: lambda x: g(f(x))
# drop :: Int -> [a] -> [a]
# drop :: Int -> String -> String
def drop(n):
'''The sublist of xs beginning at
(zero-based) index n.'''
def go(xs):
if isinstance(xs, list):
return xs[n:]
else:
take(n)(xs)
return xs
return lambda xs: go(xs)
# everyOther :: Gen [a] -> Gen [a]
def everyOther(g):
'''Every other item of a generator stream.'''
while True:
yield take(1)(g)
take(1)(g) # Consumed, not yielded.
# fst :: (a, b) -> a
def fst(tpl):
'''First component of a pair.'''
return tpl[0]
# iterate :: (a -> a) -> a -> Gen [a]
def iterate(f):
'''An infinite list of repeated applications of f to x.'''
def go(x):
v = x
while True:
yield v
v = f(v)
return lambda x: go(x)
# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.'''
return lambda xs: (
xs[0:n]
if isinstance(xs, list)
else list(islice(xs, n))
)
# uncurry :: (a -> b -> c) -> ((a, b) -> c)
def uncurry(f):
'''A function over a tuple
derived from a curried function.'''
return lambda xy: f(xy[0])(
xy[1]
)
# zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
def zipWith(f):
'''A list constructed by zipping with a
custom function, rather than with the
default tuple constructor.'''
return lambda xs: lambda ys: (
list(map(f, xs, ys))
)
# MAIN ---
if __name__ == '__main__':
main()

15
Task/Catalan-numbers-Pascals-triangle/Quackery/catalan-numbers-pascals-triangle.quackery Normal file
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[ [] 0 rot 0 join
witheach
[ tuck +
rot join swap ]
drop ] is nextline ( [ --> [ )
[ ' [ 1 ] swap times
[ nextline nextline
dup dup size 2 /
split nip
2 split drop
do - echo sp ]
drop ] is catalan ( n --> )
15 catalan

11
Task/Catalan-numbers-Pascals-triangle/REXX/catalan-numbers-pascals-triangle-1.rexx Normal file
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/*REXX program obtains and displays Catalan numbers from a Pascal's triangle. */
parse arg N . /*Obtain the optional argument from CL.*/
if N=='' | N=="," then N=15 /*Not specified? Then use the default.*/
numeric digits max(9, N%2 + N%8) /*so we can handle huge Catalan numbers*/
@.=0; @.1=1 /*stem array default; define 1st value.*/
do i=1 for N; ip=i+1
do j=i by -1 for N; jm=j-1; @.j=@.j+@.jm; end /*j*/
@.ip=@.i; do k=ip by -1 for N; km=k-1; @.k=@.k+@.km; end /*k*/
say @.ip - @.i /*display the Ith Catalan number. */
end /*i*/ /*stick a fork in it, we're all done. */

13
Task/Catalan-numbers-Pascals-triangle/REXX/catalan-numbers-pascals-triangle-2.rexx Normal file
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/*REXX program obtains and displays Catalan numbers from a Pascal's triangle. */
parse arg N . /*Obtain the optional argument from CL.*/
if N=='' | N=="," then N=15 /*Not specified? Then use the default.*/
numeric digits max(9, N%2 + N%8) /*so we can handle huge Catalan numbers*/
@.=0; @.1=1 /*stem array default; define 1st value.*/
do i=1 for N; ip=i+1
do j=i by -1 for N; @.j=@.j+@(j-1); end /*j*/
@.ip=@.i; do k=ip by -1 for N; @.k=@.k+@(k-1); end /*k*/
say @.ip - @.i /*display the Ith Catalan number. */
end /*i*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
@: parse arg !; return @.! /*return the value of @.[arg(1)] */

13
Task/Catalan-numbers-Pascals-triangle/REXX/catalan-numbers-pascals-triangle-3.rexx Normal file
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/*REXX program obtains and displays Catalan numbers from a Pascal's triangle. */
parse arg N . /*Obtain the optional argument from CL.*/
if N=='' | N=="," then N=15 /*Not specified? Then use the default.*/
numeric digits max(9, N%2 + N%8) /*so we can handle huge Catalan numbers*/
do j=1 for N /* [↓] display N Catalan numbers. */
say comb(j+j, j) % (j+1) /*display the Jth Catalan number. */
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
!: procedure; parse arg z; _=1; do j=1 for arg(1); _=_*j; end; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
comb: procedure; parse arg x,y; if x=y then return 1; if y>x then return 0
if x-y<y then y=x-y; _=1; do j=x-y+1 to x; _=_*j; end; return _/!(y)

15
Task/Catalan-numbers-Pascals-triangle/REXX/catalan-numbers-pascals-triangle-4.rexx Normal file
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/*REXX program obtains and displays Catalan numbers from a Pascal's triangle. */
parse arg N . /*Obtain the optional argument from CL.*/
if N=='' | N=="," then N=15 /*Not specified? Then use the default.*/
numeric digits max(9, N%2 + N%8) /*so we can handle huge Catalan numbers*/
!.=.
do j=1 for N /* [↓] display N Catalan numbers. */
say comb(j+j, j) % (j+1) /*display the Jth Catalan number. */
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
!: procedure expose !.; parse arg z; if !.z\==. then return !.z; _=1
do j=1 for arg(1); _=_*j; end; !.z=_; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
comb: procedure expose !.; parse arg x,y; if x=y then return 1; if y>x then return 0
if x-y<y then y=x-y; _=1; do j=x-y+1 to x; _=_*j; end; return _/!(y)

11
Task/Catalan-numbers-Pascals-triangle/Racket/catalan-numbers-pascals-triangle.rkt Normal file
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#lang racket
(define (next-half-row r)
(define r1 (for/list ([x r] [y (cdr r)]) (+ x y)))
`(,(* 2 (car r1)) ,@(for/list ([x r1] [y (cdr r1)]) (+ x y)) 1 0))
(let loop ([n 15] [r '(1 0)])
(cons (- (car r) (cadr r))
(if (zero? n) '() (loop (sub1 n) (next-half-row r)))))
;; -> '(1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900
;; 2674440 9694845)

9
Task/Catalan-numbers-Pascals-triangle/Raku/catalan-numbers-pascals-triangle.raku Normal file
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constant @pascal = [1], -> @p { [0, |@p Z+ |@p, 0] } ... *;
constant @catalan = gather for 2, 4 ... * -> $ix {
my @row := @pascal[$ix];
my $mid = +@row div 2;
take [-] @row[$mid, $mid+1]
}
.say for @catalan[^20];

13
Task/Catalan-numbers-Pascals-triangle/Ring/catalan-numbers-pascals-triangle.ring Normal file
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n=15
cat = list(n+2)
cat[1]=1
for i=1 to n
for j=i+1 to 2 step -1
cat[j]=cat[j]+cat[j-1]
next
cat[i+1]=cat[i]
for j=i+2 to 2 step -1
cat[j]=cat[j]+cat[j-1]
next
see "" + (cat[i+1]-cat[i]) + " "
next

11
Task/Catalan-numbers-Pascals-triangle/Ruby/catalan-numbers-pascals-triangle.rb Normal file
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@ -0,0 +1,11 @@
def catalan(num)
t = [0, 1] #grows as needed
(1..num).map do |i|
i.downto(1){|j| t[j] += t[j-1]}
t[i+1] = t[i]
(i+1).downto(1) {|j| t[j] += t[j-1]}
t[i+1] - t[i]
end
end
p catalan(15)

9
Task/Catalan-numbers-Pascals-triangle/Run-BASIC/catalan-numbers-pascals-triangle.basic Normal file
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n = 15
dim t(n+2)
t(1) = 1
for i = 1 to n
for j = i to 1 step -1 : t(j) = t(j) + t(j-1): next j
t(i+1) = t(i)
for j = i+1 to 1 step -1: t(j) = t(j) + t(j-1 : next j
print t(i+1) - t(i);" ";
next i

27
Task/Catalan-numbers-Pascals-triangle/Rust/catalan-numbers-pascals-triangle.rust Normal file
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fn main()
{let n=15usize;
let mut t= [0; 17];
t[1]=1;
let mut j:usize;
for i in 1..n+1
{
j=i;
loop{
if j==1{
break;
}
t[j]=t[j] + t[j-1];
j=j-1;
}
t[i+1]= t[i];
j=i+1;
loop{
if j==1{
break;
}
t[j]=t[j] + t[j-1];
j=j-1;
}
print!("{} ", t[i+1]-t[i]);
}
}

5
Task/Catalan-numbers-Pascals-triangle/Scala/catalan-numbers-pascals-triangle.scala Normal file
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def catalan(n: Int): Int =
if (n <= 1) 1
else (0 until n).map(i => catalan(i) * catalan(n - i - 1)).sum
(1 to 15).map(catalan(_))

13
Task/Catalan-numbers-Pascals-triangle/Scilab/catalan-numbers-pascals-triangle.scilab Normal file
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n=15
t=zeros(1,n+2)
t(1)=1
for i=1:n
for j=i+1:-1:2
t(j)=t(j)+t(j-1)
end
t(i+1)=t(i)
for j=i+2:-1:2
t(j)=t(j)+t(j-1)
end
disp(t(i+1)-t(i))
end

21
Task/Catalan-numbers-Pascals-triangle/Seed7/catalan-numbers-pascals-triangle.seed7 Normal file
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@ -0,0 +1,21 @@
$ include "seed7_05.s7i";
const proc: main is func
local
const integer: N is 15;
var array integer: t is [] (1) & N times 0;
var integer: i is 0;
var integer: j is 0;
begin
for i range 1 to N do
for j range i downto 2 do
t[j] +:= t[j - 1];
end for;
t[i + 1] := t[i];
for j range i + 1 downto 2 do
t[j] +:= t[j - 1];
end for;
write(t[i + 1] - t[i] <& " ");
end for;
writeln;
end func;

11
Task/Catalan-numbers-Pascals-triangle/Sidef/catalan-numbers-pascals-triangle.sidef Normal file
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func catalan(num) {
var t = [0, 1]
(1..num).map { |i|
flip(^i ).each {|j| t[j+1] += t[j] }
t[i+1] = t[i]
flip(^i.inc).each {|j| t[j+1] += t[j] }
t[i+1] - t[i]
}
}
say catalan(15).join(' ')

14
Task/Catalan-numbers-Pascals-triangle/Smart-BASIC/catalan-numbers-pascals-triangle.basic Normal file
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PRINT "Catalan Numbers from Pascal's Triangle"!PRINT
x = 15
DIM t(x+2)
t(1) = 1
FOR n = 1 TO x
FOR m = n TO 1 STEP -1
t(m) = t(m) + t(m-1)
NEXT m
t(n+1) = t(n)
FOR m = n+1 TO 1 STEP -1
t(m) = t(m) + t(m-1)
NEXT m
PRINT n,"#######":t(n+1) - t(n)
NEXT n

14
Task/Catalan-numbers-Pascals-triangle/TI-83-BASIC/catalan-numbers-pascals-triangle.basic Normal file
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"CATALAN
15→N
seq(0,I,1,N+2)→L1
1→L1(1)
For(I,1,N)
For(J,I+1,2,-1)
L1(J)+L1(J-1)→L1(J)
End
L1(I)→L1(I+1)
For(J,I+2,2,-1)
L1(J)+L1(J-1)→L1(J)
End
Disp L1(I+1)-L1(I)
End

13
Task/Catalan-numbers-Pascals-triangle/Tcl/catalan-numbers-pascals-triangle.tcl Normal file
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proc catalan n {
set result {}
array set t {0 0 1 1}
for {set i 1} {[set k $i] <= $n} {incr i} {
for {set j $i} {$j > 1} {} {incr t($j) $t([incr j -1])}
set t([incr k]) $t($i)
for {set j $k} {$j > 1} {} {incr t($j) $t([incr j -1])}
lappend result [expr {$t($k) - $t($i)}]
}
return $result
}
puts [catalan 15]

20
Task/Catalan-numbers-Pascals-triangle/VBScript/catalan-numbers-pascals-triangle.vb Normal file
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dim t()
if Wscript.arguments.count=1 then
n=Wscript.arguments.item(0)
else
n=15
end if
redim t(n+1)
't(*)=0
t(1)=1
for i=1 to n
ip=i+1
for j = i to 1 step -1
t(j)=t(j)+t(j-1)
next 'j
t(i+1)=t(i)
for j = i+1 to 1 step -1
t(j)=t(j)+t(j-1)
next 'j
Wscript.echo t(i+1)-t(i)
next 'i

11
Task/Catalan-numbers-Pascals-triangle/Vala/catalan-numbers-pascals-triangle.vala Normal file
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void main() {
const int N = 15;
uint64[] t = {0, 1};
for (int i = 1; i <= N; i++) {
for (int j = i; j > 1; j--) t[j] = t[j] + t[j - 1];
t[i + 1] = t[i];
for (int j = i + 1; j > 1; j--) t[j] = t[j] + t[j - 1];
print(@"$(t[i + 1] - t[i]) ");
}
print("\n");
}

16
Task/Catalan-numbers-Pascals-triangle/Visual-Basic/catalan-numbers-pascals-triangle.vb Normal file
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Sub catalan()
Const n = 15
Dim t(n + 2) As Long
Dim i As Integer, j As Integer
t(1) = 1
For i = 1 To n
For j = i + 1 To 2 Step -1
t(j) = t(j) + t(j - 1)
Next j
t(i + 1) = t(i)
For j = i + 2 To 2 Step -1
t(j) = t(j) + t(j - 1)
Next j
Debug.Print i, t(i + 1) - t(i)
Next i
End Sub 'catalan

10
Task/Catalan-numbers-Pascals-triangle/Wren/catalan-numbers-pascals-triangle.wren Normal file
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var n = 15
var t = List.filled(n+2, 0)
t[1] = 1
for (i in 1..n) {
if (i > 1) for (j in i..2) t[j] = t[j] + t[j-1]
t[i+1] = t[i]
if (i > 0) for (j in i+1..2) t[j] = t[j] + t[j-1]
System.write("%(t[i+1]-t[i]) ")
}
System.print()

10
Task/Catalan-numbers-Pascals-triangle/XPL0/catalan-numbers-pascals-triangle.xpl0 Normal file
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def N = 15;
int T(N+2), I, J;
[T(0):= 0; T(1):= 1;
for I:= 1 to N do
[for J:= I downto 2 do T(J):= T(J) + T(J-1);
T(I+1):= T(I);
for J:= I+1 downto 2 do T(J):= T(J) + T(J-1);
IntOut(0, T(I+1) - T(I)); ChOut(0, ^ );
];
]

9
Task/Catalan-numbers-Pascals-triangle/ZX-Spectrum-Basic/catalan-numbers-pascals-triangle.basic Normal file
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10 LET N=15
20 DIM t(N+2)
30 LET t(2)=1
40 FOR i=2 TO N+1
50 FOR j=i TO 2 STEP -1: LET t(j)=t(j)+t(j-1): NEXT j
60 LET t(i+1)=t(i)
70 FOR j=i+1 TO 2 STEP -1: LET t(j)=t(j)+t(j-1): NEXT j
80 PRINT t(i+1)-t(i);" ";
90 NEXT i

42
Task/Catalan-numbers-Pascals-triangle/Zig/catalan-numbers-pascals-triangle.zig Normal file
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const std = @import("std");
const stdout = std.io.getStdOut().outStream();
pub fn main() !void {
var n: u32 = 1;
while (n <= 15) : (n += 1) {
const row = binomial(n * 2).?;
try stdout.print("{d:2} {d:8}\n", .{ n, row[n] - row[n + 1] });
}
}
pub fn binomial(n: u32) ?[]const u64 {
if (n >= rmax)
return null
else {
const k = n * (n + 1) / 2;
return pascal[k .. k + n + 1];
}
}
const rmax = 68;
const pascal = build: {
@setEvalBranchQuota(100_000);
var coefficients: [(rmax * (rmax + 1)) / 2]u64 = undefined;
coefficients[0] = 1;
var j: u32 = 0;
var k: u32 = 1;
var n: u32 = 1;
while (n < rmax) : (n += 1) {
var prev = coefficients[j .. j + n];
var next = coefficients[k .. k + n + 1];
next[0] = 1;
var i: u32 = 1;
while (i < n) : (i += 1)
next[i] = prev[i] + prev[i - 1];
next[i] = 1;
j = k;
k += n + 1;
}
break :build coefficients;
};

2
Task/Catalan-numbers-Pascals-triangle/Zkl/catalan-numbers-pascals-triangle.zkl Normal file
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fcn binomial(n,k){ (1).reduce(k,fcn(p,i,n){ p*(n-i+1)/i },1,n) }
(1).pump(15,List,fcn(n){ binomial(2*n,n)-binomial(2*n,n+1) })