I have this recursive equation:
$$\begin{align*} F(m,n)&=F(m,n-1)+F(m-1,n)-F(m-1,n-1-m)\\\\ F(m,0)&=F\left(m,\frac12m(m+1)\right)=1\\ F(m,i)&=0\text{ if }i<0\text{ or }i>\frac12m(m+1) \end{align*}$$
is there a way to solve this recursive equation to get a close form for $F(m,n)$
I have tried z transform, but I got nothing