Compositions - Fruit Salad I'm asked to find $s(n)$ which is the number of ways to make a fruit salad with $n$ pieces of fruit, given that we must use strawberries by the half-dozen, an odd number of apples, between 2 and 7 bananas and at most 1 pineapple.
I started evaluating the ordinary generating function $S(X)$. I found
$$S(X)=\frac{1}{1-x^6}\frac{x}{1-x^2}\frac{x^2-x^8}{1-x}(1+x)=\frac{1}{1-x^6}\frac{x}{1-x}\frac{x^2-x^8}{1-x}$$
However, I don't manage to find a closed formula for $s(n)$ after. I know that I'm supposed to "evalute $S(X)$ at $x^n$" but I don't really understand how to do it. 
Any help would be appreciated.
Thanks.

You can write:
\begin{align}
S(z) &= \frac{1}{1 - z^6} \frac{z}{1 - z^2} \frac{z^2 - z^8}{1 - z} (1 + z) \\
     &= \frac{1}{1 - z^6} \frac{z}{(1 - z) (1 + z)} 
           \frac{z^2 (1 - z^6)}{1 - z} (1 + z) \\
     &= \frac{z^3}{(1 - z)^2} \\
     &= z^3 \sum_{k \ge 0} \binom{-2}{k} (-1)^k z^k \\
     &= z^3 \sum_{k \ge 0} \binom{k + 1}{1} z^k \\
     &= z^3 \sum_{k \ge 0} (k + 1) z^k
\end{align}
I'm sure you can take it from here.
