I have found a combination of generating functions where the coefficients represent a quantity I am looking for: $\frac{(1-x^5)(x-x^5)}{(1-x^{10})(1-x)^3}$.
However, I am now stuck trying to simplify this function into a series where I can extract the coefficient. I’ve tried to use partial fractions, but only one of the terms I get can be represented by a series. I’ve also tried looking to see if I can use derivatives of common series to sub in but that hasn’t gotten me anywhere either. Are there other ways to simplify functions into generating series?
Partial fraction over $\mathbb Q$ seems to be $$ \frac{2}{(-1+x)^2}+\frac{(-2x^2-x-2)}{(x^4-x^3+x^2-x+1)} $$ I assume you can do the first one. As it is, the second one (using long division) gives us a linear $5$-term recurrence for the coefficients of the power series. In general, that would be the best we can do. But in this case we can simplify. If we multiply numerator and denominator by $x+1$ we get $$ \frac{(-2x^2-x-2)}{(x^4-x^3+x^2-x+1)} = \frac{-2x^3-3x^2-3x-2}{x^5+1} $$ so we can see it as a sum of four series, each one a geometric series with ratio $-x^5$.
The result: $$ \sum_{n=0}^\infty 2(n+1)x^n + \sum_{n=0}^\infty (-1)^n (-2) x^{5n+3} + \sum_{n=0}^\infty (-1)^n (-3) x^{5n+2} \\ + \sum_{n=0}^\infty (-1)^n (-3) x^{5n+1} + \sum_{n=0}^\infty (-1)^n (-2) x^{5n} $$ So, for example, if we want the coefficient of $x^{4357}$: divide $4357$ by $5$ to see $4357 = 871\cdot 5 + 2$, so that our coefficient is $$ 2(4357+1) + (-1)^{871}(-3) = 8719 . $$