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Faulhaber's formula
Expression for sums of powers of integers
Faulhaber's formula
In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the pth powers of the first n positive integers ∑k=1nkp=1p+2p+3p+⋯+np as a polynomial in n. In modern notation, Faulhaber's formula is ∑k=1nkp=1p+1∑r=0p(p+1r)Br+np+1−r. Here, (p+1r) is the binomial coefficient "p+1 choose r", and the Br+ are the second Bernoulli numbers, identical to the first ones except for B1+=12.
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