A short proof of the Almkvist-Meurman theorem

We give a short generating function proof of the Almkvist-Meurman theorem: For integers $h$ and $k\ne0$, define the numbers $M_n(h,k)$ by $kx(e^{hx}-1)/(e^{kx}-1)=\sum_{n=0}^\infty M_n(h,k) x^n/n!$. Equivalently, $M_n(h,k) = k^n(B_n(h/k) - B_n)$, where $B_n(u)$ is the Bernoulli polynomial. Then $M_n(h,k)$ is an integer. The proof is related to Postnikov's functional equation for the generating function for intransitive trees.