TITLE: General expression of $f(a, b)$ if $f(a, b)=f(a-1,b) + f(a, b-1) + f(a-1, b-1)$?
QUESTION [5 upvotes]: $f(a,b) = f(a-1, b) + f(a-1, b-1) + f(a, b-1), ab \neq 0$ 
$f(a,b) = 1, ab = 0$ 
So what is $f(a, b)$?

REPLY [4 votes]: Following Robin Chapman's answer, $$\sum_{a,b \geq 0} f(a,b)x^a y^b = \frac{1}{1-x-y-xy} = \sum_{n \geq 0} (x+y+xy)^n = \sum_{n \geq 0} \sum_{i+j+k=n} \frac{n!}{i!j!k!} x^i y^j (xy)^k$$
so $$f(a,b)=\sum_{k=0}^{\min (a,b)} \frac{(a+b-k)!}{(a-k)!(b-k)!k!}$$