What is the solve of F(n,n) = F(n-1,n) + F(n, n-1) + 1 Where F(0,a) = 1 and F(a, 0) = 1 for every a I'm given the following python function:
def recurser(i, j):
    x = 0
    if j == 0:
        return 1
    if i == 0:
        return 1
    x += recurser(i, j - 1)
    x += recurser(i - 1, j)
    x += 1
    return x

And I'm Asked to find x for any i = j = n where n can be any positive integer. however the recursion can do the job but the question says no recursion is allowed so that I have to solve the following recursive function:

F(n,n) = F(n-1,n) + F(n, n-1) + 1
F(0,a) = 1 for every positive a
F(a, 0) = 1 for every positive a

Is there any solution for it?
Your function produces the sequence A109128, that is
$$
\mathit{recurser}(i,j) = 2\binom{i+j}{i} - 1.
$$
You can prove this by induction.
How I found out: I computed the first few values, and searched the OEIS.