Sums of powers of integers and the sequence A304330

For integer k ≥ 1, let Sₖ(n) denote the sum of the kth powers of the first n positive integers 1ᵏ + 2ᵏ + ⋯ + nᵏ. In this paper, we derive a new formula expressing 2²ᵏ times S₂ₖ(n) as a sum of k terms involving the numbers in the kth 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 S₂ₖ(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 S₂ₖ₋₁(n). Finally, we determine the Faulhaber form of S₂ₖ(n) and S₂ₖ₊₁(n) in terms of the sequence A304330 and the Legendre-Stirling numbers of the first kind.