partial alternating sum involving binomial coefficients

In a post on Math Stack Exchange, MSE 2827591, I prove the following:

$$\sum_{k=1}^n (-1)^{k+1} \binom{2n}{n+k} k^s = \binom{2n}{n} \sin(\pi s/2) \int_0^\infty \frac{dx \, \,x^s}{\sinh{\pi x}} \frac{n!^2}{(n+ix)!(n-ix)!}. $$

The OP's formula can be put in the form on the LHS of this equation. By inspection the questions of concern can be answered; namely, the value of zero for $s$ an even integer, and non-zero otherwise.