For integers k ≥ 1, let S_k(n) denote the sum of the kth 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 kth 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-1S_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.