While solving a certain physical problem, I have reached the following summation:
$$S = \sum_{k = 0}^n \left({n-k-\frac{1}{2} \choose n-k}\right)^2 \ \ \frac{(k+l)!}{k!}$$
where $n, k$ and $l$ are all non-negative integers. The sum over $k$ (as can be seen from the above equation) has to be computed for specified $(n, l)$ values.
While it is possible to evaluate this summation for specified $n$ and $l$ values, using software packages, I was wondering if there is any way to use some $n \choose k$ and factorial properties to analytically reduce this expression to a simpler form, ideally even a closed form expression in terms of $n$ and $l$.
Is that possible, or is this already the simplest possible form of this summation, not reducible any further?
Maple writes this using a hypergeometric function:
$$ \left( {n-1/2\choose n} \right) ^{2}l!\, {\mbox{$_3$F$_2$}(-n,-n,l+1;\,-n+1/2,-n+1/2;\,1)} $$