Specific recurrence in $2$ variables

Question

I have the following relation:

$$F(x, y) = F(x, y - 1) + F(x - 1, y - 1)$$

and the initial conditions: $F(x, 1) = 1$ and $F(1, y) = y$.

How can I solve this recurrence?

Thank you in advance!

Answer 1

I have programmed the recursive relation and obtained the result below, the blue lines indicating the "computation flow", for example on North-West corner: $46 = 9 + 37$.

You should be able, from this table, to elaborate conjectures (for example about powers of 2 that you can see on and below line with equation $y=x+1$).