 Faulhaber polynomial

For a fixed nonnegative integer $p$, the Faulhaber polynomial $S_p(n)$ is the rational-coefficient polynomial satisfying $S_p(n)={\sum }_{k=1}^n{k}^p$ for nonnegative integers $n$. It is characterized by $S_p(0)=0$ and $S_p(n)-S_p(n-1)=n^p$.

For example, $S_2(n)=n(n+1)(2n+1)/6$ has degree three and leading coefficient $1/3$. In general the degree is $p+1$ and the leading coefficient is $1/(p+1)$, since taking a finite difference lowers the degree by one. Faulhaber's formula constructs these polynomials from the Bernoulli polynomials.