I have a generating function
$$\sum_{m,n} P(m,n)y^{m}x^{n} = \prod(1-yx^t)^{-1}$$
where $t \in \{1,2,3,5,8,...\}$ i.e. Fibonacci set with single 1.
For now, I have a naive script to calculate coefficient of $y^{m}x^{n}$ which works by expanding and ignoring higher order terms but this gets slow with $n$ greater than $10^{6}$
I saw various coefficient derivations for generating functions in one variable, however I am unable to get a clean form in terms of combinations for the above generating function.
Is it possible to get a function(m,n) for the coefficients?
EDIT : The given generating function calculates how many ways to add up to n using exactly m numbers from the set