Is there a sharper bound than exponential for $\sum_{k\ge0}\frac{m!(k+n-m)!}{(k+n)!}\frac{s^k}{k!}$?

Question

I am trying find a bound for an expression and I am getting something not quite as convenient as I hoped. Going through my calculations again I think that the only place I use a non sharp bound is when I wrote $$\sum_{k\ge0}\frac{1}{\binom{k+n}{m}}\frac{s^k}{k!}\le e^s,$$ for $m\le n$, which of course comes from the fact that $\binom{k+n}{m}\ge1$. So, is there a sharper bound for that series or I am stuck with it?