I want to find the coefficient of $x^n$ in $G(x)$ where
$ G(x) = \frac{1}{1-x^{a_1}}\times\frac{1}{1-x^{a_2}}\times\dots\times\frac{1}{1-x^{a_k}}$
how do I approach this? It would be helpful if it is explained using a specific example with at least $3$ fraction terms on RHS
Note: I am looking for standard approach.
The coefficient of $x^n$ in $\frac1{1-x}$ is $1$. The coefficient of $x^n$ in $\frac1{1-x^a}$ is $1$ if $n$ ia a multiple of $a$, $0$ otherwise; in other words: it is the number of ways to partition $n$ into parts of size $a$. From the way the Cauchy product of power series work, this last interpretation can be generalized to your $G$: The coefficient of $x^n$ is the number of ways to partition $n$ into parts of size $a_1, a_2, \ldots, a_k$ (assuming all $a_i$ different for simplicity)