<s>[INST] Transform $\prod_{k=0}^{n} (1+x^{2^{k}})$ to $\sum_{k=0}^{n} c_{k}x^{k}$ I want to transform the following 
$$\prod_{k=0}^{n} (1+x^{2^{k}})$$
to the canonical form $\sum_{k=0}^{n} c_{k}x^{k}$
This is what I got so far
\begin{align*}
\prod_{k=0}^{n} (1+x^{2^{k}})= \dfrac{x^{2^{n}}-1}{x-1} (x^{2^{n}}+1) \\
\end{align*}
but I don't know how to continue, can someone help me with this?
 [/INST] Note that $ \sum_{k=0}^{n-1} 2^k=2^n-1$. Therefore, the highest power of $x$ in $\prod_{k=0}^{n-1}(1+x^{2^k})$ is $2^n-1$. In other words,  none of the terms will repeat as we keep multiplying by $1+x^{2^n}$. Thus, the relation can be obtained using induction. 
It is easy to see that for $n=1$
\begin{equation}
\prod_{k=0}^1(1+x^{2^k}) = (1+x)(1+x^2)= 1+x+x^2+x^3.
\end{equation}
Now, let us assume that $\prod_{k=0}^{n-1}(1+x^{2^k}) = \sum_{k=0}^{2^n-1} x^k$. Then, we have
\begin{equation}
\prod_{k=0}^{n}(1+x^{2^k}) = \left(\sum_{k=0}^{2^{n-1}} x^k\right)(1+x^{2^{n}}) = \sum_{k=0}^{2^{n}-1} x^k+ \sum_{k=0}^{2^{n}-1} x^{k+2^n} = \sum_{k=0}^{2^{n+1}-1} x^k.
\end{equation}
</s>