= Faulhaber's formula
{c}
= Faulhaber's theorem
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{synonym}
For an <integer> $p\ge0$, <Faulhaber's formula> expresses a sum of powers as a <polynomial> in the upper limit:
$$
\sum_{k=1}^{n}k^p=\frac{B_{p+1}(n+1)-B_{p+1}(1)}{p+1},
$$
where $B_j(x)$ are the <Bernoulli polynomials>. The identity $B_{p+1}(x+1)-B_{p+1}(x)=(p+1)x^p$ makes the sum telescope. This is a discrete analogue of finding an <antiderivative>.<NIST Digital Library of Mathematical Functions>[§24.4, difference equations and sums of powers]{ref}
For $p=1$ the result is $n(n+1)/2$; for $p=3$ it is $n^2(n+1)^2/4$. Although the displayed <polynomial> has rational coefficients, it takes <integer> values at nonnegative <integers>.
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