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.