Given the following recursion:
$$ F(n,d) = F(n-1,d) + F(n-1,d-1) + 1 $$
With initial conditions $F(0,d)=1,F(n,1)=1$ and $n\in\mathbb N_0, d\in\mathbb N$.
I noticed that it holds (By writing out the table for $n,d$ and doing some tweaking):
Can this solution be justified (proven) by a combinatorial argument?
I want to avoid proving it by solving the above recursion in two vairables.
The recursion seems similar to the one for diagonals in Pascals triangle.