How would I computationally find a generating functions coefficient?

Question

More specifically $a_n=(1,5,10,25,100,500,1000,2000,10000)$

$G(x)=\Pi_{n=0}^8 \sum_{i=0}^{\infty}x^{a_ni}$

So when $a_n=1$ the series = $1+x+x^2+x^3+...$

$a_n=5, 1+x^5+x^{10}+x^{15}+...$

$a_n=10, 1+x^{10}+x^{20}+x^{30}+...$

etc.

Answer 1

Here is a start.

You want $G(x)=\Pi_{n=0}^8 \sum_{i=0}^{\infty}x^{a_ni}$, so look at the sums, which are just geometric series.

$\sum_{i=0}^{\infty}x^{a_ni} =\sum_{i=0}^{\infty}(x^{a_n})^{i} =\dfrac{1}{1-x^{a_n}} $

so $G(x) =\Pi_{n=0}^8 \dfrac{1}{1-x^{a_n}} = \dfrac{1}{\Pi_{n=0}^8(1-x^{a_n})} $.

At this point, I'd probably expand into partial fractions or use logarithmic differentiation to get something that might enable the isolation of coefficients.

Sort of reminds me of Hardy and Ramanujan's stuff.