<s>[INST] Generating Functions Partitions Let $U(x)=\sum_{n=0}^{\infty} u_nx^n$, where $u_n$ is the number of partitions of $n$ into at most two parts. For example, $u_4=3$ because $4$ can be partitioned into at most two parts as $4$, $3+1$, or $2+2$. Use the convention that $u_0=1$.
Then $\frac 1{U(x)}$ is a polynomial. What polynomial is it? 

Can I use generating functions to solve this? If so, how?
 [/INST] Hint: It is pretty easy to see that
$$
u_n=\left\lfloor\frac n2\right\rfloor+1
$$
This will be a useful series here
$$
\begin{align}
\frac1{(1-x)^2}
&=\frac{\mathrm{d}}{\mathrm{d}x}\frac1{1-x}\\
&=\frac{\mathrm{d}}{\mathrm{d}x}\left(1+x+x^2+x^3+\dots\,\right)\\[6pt]
&=1+2x+3x^2+\dots
\end{align}
$$
Thus,
$$
\begin{align}
U(x)
&=1+x+2x^2+2x^3+3x^4+3x^5+\cdots\\[12pt]
&=(1+x)(1+2x^2+3x^4+\cdots\,)\\[6pt]
&=\frac{1+x}{(1-x^2)^2}
\end{align}
$$
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