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?

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