How would I computationally find a generating functions coefficient? 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.

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.
