= Faulhaber polynomial
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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>.
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