If the generating function is (1-x^200)(1-x^240)/((1-x^10)(1-x^20)(1-x^25)). Then expand all the terms through x^100. How can I make it in the form of a recurrence relation hn. The initial conditions are given.
The generating function is
$\displaystyle (1-x^{200})(1-x^{240})(1-x^{10})^{-1}(1-x^{20})^{-1}(1-x^{25})^{-1}$
$\displaystyle =(1-x^{200})(1-x^{240})\sum_{i=0}^\infty x^{10i} \sum_{j=0}^\infty x^{20j} \sum_{k=0}^\infty x^{25k}$
The first two factors, $\displaystyle =(1-x^{200})(1-x^{240})$, don't make any contribution to $\displaystyle x^n$ with $\displaystyle n < 200$, so all you have to do is expand enough of
$\displaystyle \sum_{i=0}^\infty x^{10i} \sum_{j=0}^\infty x^{20j} \sum_{k=0}^\infty x^{25k}$
to get the powers of $\displaystyle x$ up to $\displaystyle x^{100}$.
I would start by expanding
$\displaystyle \sum_{j=0}^5 x^{20j} \sum_{k=0}^4 x^{25k}$
up to $\displaystyle x^{100}$.
I don't know enough about what you want in the way of a recurrence to help you with that part of the problem.