Question: Need to test a recursion

I have a multivariable function, F(n, g(1,0),g(0,1),g(0,2),g(1,1),g(2,1),...,g(n,1),g(n-1,2),...,g(1,n)), of indeterminates g(i,j) (omitting g(0,0) - in other words - let L = set of all pairs of nonnegative integers, (i,j), which satsify 1<= i+j <=n) of the following form F = product over all the (i,j) in L of g(i,j)^h(i,j) / ((i! * j!)^h(i,j) * h(i,j)!)  times another function of just the exponents h(i,j) which I wish to keep secret at the moment - then summed over all functions h(i,j) in the set H, where H = the set of nonnegative integer-valued functions h(i,j) which satisfy sum over all (i,j) in L of h(i,j) = 2n-1, sum over all (i,j) in L of j*h(i,j) = n, sum over all (i,j) in L of i*h(i,j) = 2*(n-1).

The product of the g(i,j)^h(i,j), factorial functions etc over all (i,j) in L is trivial. No conditionals are involved, because I can take i = 1 to n and j = 1 to n and when i+j goes out of bounds, i..e when i+j>n, the conditions on H will make h(i,j)=0.

But, how do I sum over all h(i,j) constrained to the set H?

I face this problem everywhere: having to sum over vectors (whose components are indexed by L) constrained by some (usually linear) equations. But, I have no idea how to set up the conditionals, and then transfer that to my summations.

In case you've not figured out, I am doing something very similar to (but not exactly) Bell polynomials.