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 not guarantees that this is correct (especially as the original answer contained a $\binom{n^2}{-1}$ in it).

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).