The recurrence relation is: $F(1,1)=1, F(n,k)=0$ if $k \leq 0$, $F(n,k)=0,$ if $k \geq 2n$ where $$F(n+1,k) = F(n,k-2)F(n,k-1) + F(n,k).$$ I'm looking for a closed form solution for $F(n,n)$. Else, if not possible finding a generating function for it?
I have tried generating functions, but the first term is troublesome.