This is problem 2 from Polya's Problems and Theorems in Analysis I. The question is as follows,
Let $A_n$ denote the number of solutions to the Diophantine equation $x+5y+10z+25u+50v=n$. What is the rational function of $\zeta$ represented by the series $A_0+A_1\zeta+A_2\zeta^2+\dots+A_n\zeta^n+\dots$ ?
I own the book so I have the solution in the back, but how would I derive it? Is there a place on the internet where I can familiarize myself with infinite series that correspond to the solutions of Diophantine equations?
Let's work out a smaller example. Let $B_n$ represent the number of solutions of $x + 5y = n$. Consider
$$ f(\zeta) = (1 + \zeta + \zeta^2 + ...)(1 + (\zeta^5) + (\zeta^5)^2 + ...)$$
Perform the multiplication to see that a coefficient at $\zeta^n$ is $B_n$. Calculate both factors as geometrical series to obtain a rational function. Apply the same logic to an original equation.