How to find the coefficient of a power in a power series?? How can I find the coefficient of $x^{80}$ in the power series $$(1+x+x^{2}+x^{3}+x^{4}+\cdots)(x^{2}+x^{4}+x^{6}+x^{8}+\cdots)(1+x^{3}+x^{5})\,?$$
Is there a general method to this? 

We have
$(1+x+x^2+\cdots)(x^2+x^4+\cdots)(1+x^3+x^5)$
Then we want The coeficient of $x^{80}$, note that $x^nx^m=x^{n+m}$ then searching The ways of write $80$ using the sets $\{0,1,\cdots\},\{2,4,\cdots\},\{0,3,5\}$.
$$\begin{align}
80&=0+80+0\\
&=2+78+0\\
&=4+76+0\\
&\vdots\\
&=78+2+0\\
&=1+76+3\\
&=3+74+3\\
&=5+72+3\\
&\vdots\\
&=75+2+3\\
&=1+74+5\\
&=3+72+5\\
&\vdots\\
&=73+2+5
\end{align}$$
Then The coeficiente will be
$$\begin{align}
C_{80}&=\stackrel{2n}{(40-1+1)}+\stackrel{2n+1}{(37-0+1)}+\stackrel{2n+1}{(36-0+1)}\\
&=40+38+37\\
&=115
\end{align}$$
Then if nothing as wrong The answer must be $115$
