Summing series with factorials in 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)

Try maxima on this. Load zeilberger, and:
GosperSum(y * binomial(2 m, m + y), y, 1, m)

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