Let $f(n)$ be the number of ways to write $n$ as sum of positive odd integers that each one of them is less than $10$, without any importance to their order. For example: $f(6)=4$ as you can write it as $1+1+1+1+1+1,5+1,3+3,3+1+1+1$. Find a generating function for the series ${f(n)}$.
I've tried finding a recursive relation for $f(n)$ but I got stuck.
Thanks
Best Answer
Each partition can be uniquely identified by how many copies of each number it uses. Hence, we get the generating function:
$$\left(1+x+x^2+\dots\right)\left(1+x^3+x^6+\dots\right)\cdots\left(1+x^9+x^{18}+\dots\right)=\frac{1}{\left(1-x\right)\left(1-x^3\right)\cdots\left(1-x^9\right)}$$