Summing series with factorials in

Question

How do you sum this series? $$\sum _{y=1}^m \frac{y}{(m-y)!(m+y)!}$$

My attempt:

$$\frac{y}{(m-y)!(m+y)!}=\frac{y}{(2m)!}{2m\choose m+y}$$

My thoughts were, sum this from zero, get a trivial answer, take away the first term. But actually I don't think this will work very well.

This question was originally under probability, but the problem is that I can't sum a series and really has nothing to do with probability (reason for the first comment)

Answer 1

For example, one can write

\begin{align} \sum_{y=0}^m\frac{y}{(m-y)!(m+y)!} &= \sum_{k=0}^m\frac{m-k}{k!(2m-k)!} \\ &=\frac{m}{(2m)!}\sum_{k=0}^m{2m \choose k}-\frac{1}{(2m-1)!}\sum_{k=1}^{m}{2m-1\choose k-1} \\ &= \frac{m}{2(2m)!}\left[{2m\choose m}+\sum_{k=0}^{2m}{2m \choose k}\right]-\frac{1}{(2m-1)!}\sum_{k=0}^{m-1}{2m-1\choose k} \\ &= \frac{m}{2(2m)!}\left[{2m\choose m}+\left(1+1\right)^{2m}\right]-\frac{1}{2(2m-1)!}\sum_{k=0}^{2m-1}{2m-1\choose k} \\ &= \frac{m}{2(2m)!}{2m\choose m}+\frac{m\cdot 2^{2m}}{2(2m)!}-\frac{2^{2m-1}}{2(2m-1)!} \\ &= \frac{m}{2(2m)!}{2m\choose m}\;. \end{align} All we have used in the way is that $\displaystyle{n\choose k} ={n\choose n-k}$ and that $\displaystyle(1+1)^n=\sum_{k=0}^n{n\choose k}$.

Answer 2

Try maxima on this. Load zeilberger, and:

GosperSum(y * binomial(2 m, m + y), y, 1, m)

gives $$\dfrac{1}{2}(m + 1) \binom{2 m}{m + 1}$$