I'm trying to solve the following cominatorics problem:
How many ways are there to choose $3n$ subset with repetitions allowed from set $\{A,B,C\}$ where $A, B$ are present at most $2n$ times each and $C$ is present odd times?
I think we're looking at coefficient of $t^{3n}$ in:
$(1+t+...+t^{2n})(1+t+...+t^{2n})(t+t^3+...+t^{2k+1})$ where picking $t^k$ from first bracket corresponds to picking $A$ $k$ times, same for second bracket and $B$, and third bracket and $C$.
So apparently all we need to do now is unfolding $(\frac{1-t^{2(n+1)}}{1-t})^2 * \frac{t}{1-t^2}$ into series and checking the coefficient of $t^{3n}$. Is my reasoning correct?