Summation over specific tuples

Question

I would like to write a function that would, given a positive integer $n$, compute the following:

$S_n = \sum_{(x,y,z)} a_{(x,y,z)} f(x,y,z)$, where the sum runs over all tuples $(x,y,z)$ such that $x+y+z = n$, and $x,y,z$ take values in {$0,1,...,n$}. The ultimate goal is to pass this function trough Solve to determine the coefficients $a_{(x,y,z)}$ with some other conditions from the problem.

My main problem is generating the appropriate tuples in a way that I could then pass through Sum.

Any help is appreciated.

Answer 1

As suggested by @MarcoB, here is my solution, using FrobeniusSolve:

Frobenius[n_] := FrobeniusSolve[{1, 1, 1}, n];
Sol[n_] :=
Sum[a[Part[Frobenius[n], i] /. List -> Sequence]
*OrderN[Part[Frobenius[n], i] /. List -> Sequence],
{i, 1, Length[Frobenius[n]]}]

In the code, corresponds to the $f(x,y,z)$ in the original question.