sum of product of inverse factorials for all integers $m,n$ that sum to $k$

Question

I'm wondering if there is a way to write the following in closed form in terms of the integer $k$ $$\sum_{\{m,n\}|m+n=k} \frac{1}{n!(m+1)!}$$ where, in words, I am summing up $\frac{1}{n!(m+1)!}$ for all non-negative integers $m,n$ such that their sum is $k$.

Any help appreciated! Not homework!

Answer 1

HINT: Multiply your sum by $(k+1)!$ to get

$$\sum_{n=0}^{k+1}\binom{k+1}n$$

and evaluate that first. If you really mean that $m$ must be non-negative, you’ll have to change the upper limit to $k$.