Integer ratios of consecutive alternating power sums

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