Constant term of zeta binomials

Question

Let's have the following zeta binomial $\sum\limits_{n=1}^\infty (1/n-1/(n+1))^k$, where $k$ a natural number and $k>1$. From the expansion of these binomials we obtain polynomials of $\pi$ where one of the terms is always an integer. Does anyone know how to calculate this integer, for all values of $k$, without expanding the zeta binomial?