Consider the the general linear diophantine equation $$\sum_{i=1}^{k}a_ix_i =n $$ with $a_i\geq 1, n\geq 1$ and $x_i\geq 0.$
Then the generating function that counts the number of solutions to this equation is $$F(x)=\prod_{i=1}^{k}\left(\frac{1}{1-x^{a_i}}\right) = 1+\sum_{n\geq 1}a(n)x^{n},$$ where $a(n)$ is the number of solutions to the linear Diophantine equation. Due to Schur we have that if $\gcd(a_1,a_2,\cdots,a_k)=1$ then $$a(n)\sim \frac{n^{k−1}}{(k − 1)!a_1a_2 ··· a_k}$$ as $n\to \infty.$
However, I am curious to know if there is a general asymptotic expression for $a(n)$ with perhaps no restrictions on the coefficients $a_i$ of the equation. I tried to see if I could mimic the proof of Schur's theorem to come up with something, but because in the general case we do not know which pole of $F$ has the highest multiplicity, it is hard to guess the answer.
Any ideas in this regard will be much appreciated.