An algorithm for the Faulhaber polynomials

Abstract

Let $S_p(n)$ denote the sum of $p$th powers of the first $n$ positive integers $1^p + 2^p + \cdots + n^p$. In this paper, first we express $S_p(n)$ in the so-called Faulhaber form, namely, as an even or odd polynomial in $(n + 1/2)$, according as $p$ is odd or even. Then, using the relation $S_p(n) - S_p(n-1) = n^p$, we derive a recursive formula for the associated Faulhaber coefficients. Applying Cramer's rule to the corresponding system of equations, we obtain an explicit determinant formula for the said coefficients. Furthermore, we show how to convert the (even or odd) Faulhaber polynomials in $(n+ 1/2)$ into polynomials in $S_1(n)$ for any arbitrary $p$, and vice versa.