Dear mathematicians,

I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In doing so, I designed the following double summation:
$$
\sum_{h = 1}^{n}\sum_{j=0}^{n-h}h{2j+h-1\choose j}{2n - 2j - h\choose n-j},
$$
which, divided by the $n$th Catalan number, yields the statistics I want. Since the variable $j$ occurs in many places, I thought that I could exchange the summations without worrying about the bounds and then work on $h$ instead:
$$
\sum_{j\geq 0}\sum_{h>0}h{(2j-1)+h\choose j}{2(n-j)-h\choose n -j}.
$$
Does anyone know if there is a closed form for
$$
\sum_{h>0}{h}{p+h\choose q}{2r - h \choose r}?
$$
I read the relevant chapter in *Concrete Mathematics*, to no avail. Perhaps generating functions would help? Anyhow, I would be interested in the main term of the asymptotic expansion of the original double summation.

Thanks for reading me.

A=Bby Petkovsek, Wilf, and Zeilberger. – Steve Huntsman Aug 30 '11 at 13:32