 Faulhaber's formula

For an integer $p\ge 0$, Faulhaber's formula expresses a sum of powers as a polynomial in the upper limit: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.^[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.