Simplify Math Expression

Prompt

Given: F(1,1)=1,\quad F(i,j)=\frac{1}{2}\sum_{k=0}^{\min(i-1,j-1)}\binom{i-1}{k}\binom{j-1}{k}2^{i+j-2-k}\quad\text{for }(i,j)\neq(1,1). \] G(i,j)=F(n+1-i,\;n+1-j). T(n)=F(n,n)+\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{p=1}^{i}\sum_{q=1}^{j} F(i,j)\,G(p,q)\;-\;\sum_{i=1}^{n}\sum_{j=1}^{n} F(i,j)\,G(i,j), \] Simplify T(n) as much as possible. Reduce it to its simplest form.