It is given that $T(i,j) = T(i-1,j) + T(i,j-1)$ with boundary conditions $T(i,0)=T(0,j)=1$. Does there exist a closed-form mathematical expression for $T(n,n)$?
I tried making a table of values of $T(i,j)$ for $i,j=1,\ldots,n$. Each entry is the sum of values to its top and left. So if we fill the rowwise, we will eventually get to $T(n,n)$. But that leads me nowhere near a closed-form expression.
Is there some other method that gives a closed-form expression?
Hint.
We can try the generating function technique. Calling
$$ T(x,y) = \sum_{i\ge 0}\sum_{j \ge 0}t(i,j)x^i y^j $$
we have
$$ T(x,y) = x\left(T(x,y)-\sum_{j\ge 0}y^j\right)+y\left(T(x,y)-\sum_{i\ge 0}x^i\right) $$
or considering $|x| < 1, |y| < 1$
$$ T(x,y) =\frac{1}{1-x}+\frac{1}{1-y}- \frac{2}{1-x-y} $$