A short proof of the Almkvist-Meurman theorem

We give a short generating function proof of the Almkvist-Meurman theorem: For integers $h$ and $k\mathrm{\ne }0$, define the numbers $M_n(h,k)$ by $kx(e^{hx}-1)\mathrm{/}(e^{kx}-1)={\sum }_{n=0}^{\mathrm{\infty }}{M}_n(h,k)x^n\mathrm{/}n!$. Equivalently, $M_n(h,k)=k^n(B_n(h\mathrm{/}k)-B_n)$, 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.