Finding coefficient of generating functions I have the equation 
$$(1+x+x^2+\ldots+x^k+\ldots)(1+x^2+x^4+\ldots+x^{2k}+\ldots)(x^2+x^3)$$ 
how of I find the coefficent of $x^{24}$. I know to condense this down to 
$$\frac1{1-x}\cdot\frac1{1-x^2}\cdot x(1+x)$$
but I don't know what to do after that

Here's another way: the coefficient of $x^{24}$ is the number of integer solutions to $$a+b+c=24$$ where $a\ge0$; $b\ge0$ is even; and $c$ is $2$ or $3$. 
If $c=2$, then we have $a+b=22$. There are $12$ values of $b$ (namely, $0,2,\dots,22$), and for each, a value of $a$. So, $12$ solutions so far. 
If $c=3$, then $a+b=21$, so there are $11$ values of $b$ ($0,2,\dots,20$); for each, there's a value of $a$, so, another $11$ solutions. 
All told, $12+11=23$ solutions, so that's the coefficient you're looking for. 
