  For integer $k \geq 1$, let $S_k(n)$ denote the sum of the $k$th powers of
the first $n$ positive integers $1^k + 2^k + \cdots + n^k$. 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 of the second kind
with even indices. 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. We also deduce the corresponding formulas for the odd-indexed power
sums $S_{2k-1}(n)$. Finally, we determine the Faulhaber form of $S_{2k}(n)$ and
$S_{2k+1}(n)$ in terms of the sequence A304330 and the Legendre-Stirling
numbers of the first kind.
