Integer ratios of consecutive alternating power sums

Abstract: We give a characterization of all pairs $(k,n)$ of positive integers for which the ratio $$ \frac{1^{k-2k+3k-\dots+(-1){n+1}} n^{k}{1k-2k+3k-\dots+(-1){n}(n-1)k}} $$ of two consecutive alternating power sums is an integer.