Sums of powers of integers and the sequence A304330
José L. Cereceda
TL;DR
The paper develops new explicit representations for sums of powers of integers by connecting the power-sum polynomials $S_k(n)$ to the OEIS sequence A304330 through coefficients $R(k,m)$ that arise from central factorial numbers with even indices. It proves a main formula $2^{2k} S_{2k}(n)=\sum_{m=1}^k R(k,m)\binom{2n+m+1}{2m+1}$ and derives companion expressions for $S_{2k}(n)$, $S_{2k-1}(n)$, $T_{2k}(n)$, and $\Omega_{2k}(n)$, all in binomial-sum forms tied to $R(k,m)$. The work also recasts these sums in Faulhaber form using Legendre-Stirling numbers, yielding explicit coefficient formulas $b_{k,r}$, $c_{k,r}$, and $d_{k,r}$, and relates them to Knuth's classical formulas and Bernoulli numbers. Together, these results illuminate the structure of even- and odd-power sums and reveal a tight link between classical power-sum identities and A304330-centered combinatorial arrays. The findings provide a unified framework for representing power sums via binomial bases and known special-number systems, with potential applications in symbolic summation and number theory.
Abstract
For integer $k \geq 1$, let $S_k(n)$ denote the sum of the $k$th powers of the first $n$ positive integers. In this paper, we derive a new formula expressing $2^{2k}$ times $S_{2k}(n)$ as a sum of $k$ terms involving the numbers in the $k$th row of the integer sequence A304330, which is closely related to the central factorial numbers with even indices of the second kind. Furthermore, we provide an alternative proof of Knuth's formula for $S_{2k}(n)$ and show that it can equally be expressed in terms of A304330. Moreover, we obtain corresponding formulas for $2^{2k-1}S_{2k-1}(n)$ and determine the Faulhaber form of both $S_{2k}(n)$ and $S_{2k+1}(n)$ in terms of A304330 and the Legendre-Stirling numbers of the first kind.