Sums of powers of integers and the sequence A304330

TL;DR

This paper introduces new formulas connecting sums of powers of integers to the sequence A304330, involving central factorial numbers, and provides alternative proofs and Faulhaber formulas for these sums.

Contribution

It derives novel formulas expressing sums of powers in terms of sequence A304330 and offers alternative proofs and Faulhaber formulas involving special number sequences.

Findings

New formulas linking sums of powers to sequence A304330

Alternative proof of Knuth's formula for sums of powers

Faulhaber formulas expressed via A304330 and Legendre-Stirling numbers

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.