37 lines
1.1 KiB
Text
37 lines
1.1 KiB
Text
Named after [https://en.wikipedia.org/wiki/Johann_Faulhaber Johann Faulhaber], the rows of Faulhaber's triangle are the coefficients of polynomials that represent sums of integer powers, which are extracted from Faulhaber's formula:
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:<math>\sum_{k=1}^n k^p = {1 \over p+1} \sum_{j=0}^p {p+1 \choose j} B_j n^{p+1-j}</math>
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where <math>B_n</math> is the nth-Bernoulli number.
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The first 5 rows of Faulhaber's triangle, are:
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<pre>
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1
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1/2 1/2
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1/6 1/2 1/3
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0 1/4 1/2 1/4
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-1/30 0 1/3 1/2 1/5
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</pre>
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Using the third row of the triangle, we have:
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<math>\sum_{k=1}^n k^2 = {1 \over 6} n + {1 \over 2} n^2 + {1 \over 3} n^3</math>
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; Task
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:* show the first 10 rows of Faulhaber's triangle.
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:* using the 18th row of Faulhaber's triangle, compute the sum: <math>\sum_{k=1}^{1000} k^{17}</math> (extra credit).
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; See also:
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* [[Bernoulli numbers]]
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* [[Evaluate binomial coefficients]]
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* [https://en.wikipedia.org/wiki/Faulhaber%27s_formula Faulhaber's formula (Wikipedia)]
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* [http://www.ww.ingeniousmathstat.org/sites/default/files/Torabi-Dashti-CMJ-2011.pdf Faulhaber's triangle (PDF)]
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<br>
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