Dealing with Generating Functions accurately I'm currently working through Iven Niven's "Mathematics of Choice." In the chapter on Generating Functions, the exercises include problems like:
How many solutions in non-negative integers does the equation $2x+3y+7z+9r=20$ have?
This of course ends up being the same as finding the coefficient of $x^{20}$ in $(1+x^2+x^4+x^6+\cdots+x^{20})(1+x^3+x^6+\cdots+x^{18})(1+x^7+x^{14})(1+x^9+x^{18})$. Maybe I'm just being a big weenie, but dealing with polynomials this large ends up being really tedious and error-prone. 
I've realized that you can save the most involved multiplication for last, since that requires the least detail, and I'm of course not bothering to keep track of any powers greater than the one I'm interested in. 
Any tips for keeping these giant polynomials manageable?

Dealing with coefficients of generating functions is either synthetically (by learning patterns from experience what the formula for coefficients might be) or directly by manipulating the coefficients of constituent gfs (which helps with the first one). Usually the latter ends up by resulting in the multiplication of polymonials.
For multiplying large complicated polynomials in order to get a single coefficient:


*

*throw away anything above the degree you care about.

*use grid paper to manage coefficients in a matrix

*shift successive lines so you can add by columns instead of diagonals.
For example, for  $(1 + 3x^2 + x^4 + 2x^6)(1 + 5x^3 + x^6)$:
$$
\begin{array}{c|cccccccc}
\hline
    & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline
    & 1 & 0 & 3 & 0 & 1 & 0 & 2 \\
\hline
 1  & 1 &   & 3 &   & 1 &   & 2 \\
 0  &   &   &   &   &   &   &   \\
 0  &   &   &   &   &   &   &   \\
 5  &   &   &   & 5 &   & 15&   & 5 &   & 10\\
 0  &   &   &   &   &   &   &   \\
 0  &   &   &   &   &   &   &   \\
 1  &   &   &   &   &   &   & 1 &   & 3 &   & 1 &   & 2 \\
\hline
    & 1 & 0 & 3 & 5 & 1 & 15& 3 & 5 & 3 & 10& 1 & 0 & 2
\end{array}
$$
for $1 + x^2 + 5x^3 + x^4 + 15 x^5 + 3 x^6 + ...$
If all we wanted was the coefficient of $x^6$, we could have avoided any column beyond 6.
