I would like to know a closed formula for
$\sum_{j=0}^{p-n } (-1)^j\binom{n^2}{p-n-j}
\binom{n+j-1}j\binom{2n+j}{n+j+1}$, especially in the
case $p$ is near $n^2/2$. Similarly, I would like a closed formula for:
setting $q=2\cdot\lceil\frac{n(n+1)}{4}\rceil -1$,
and setting
$p=\lceil\frac{q}{2}\rceil-1$,
what is the sum
$
\sum_{j=0}^{p-n } (-1)^j\binom{q}{p-n-j}
\binom{n+j-1}j\binom{2n+j}{n+j+1}
$?
In either case I would be happy for an estimate of the growth of the
sum (divided by $\binom {n^2-1}p$ in the first case, and divided by
$\binom{q-1}p$ in the second).
Best Answer
I played around with your sum in Maple and got
$$ \frac{2n}{n+1}{2n-1 \choose n-1}{n^2 \choose p-n} 3F_{2}([n,n-p,2n+1],[n+2,n^2+n+1-p],1) $$
I make no guarantees that this is correct (especially as the original answer contained a $\binom{n^2}{-1}$ in it).