232. Power series. Expand $\frac{1}{(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)}$ into a power series.
232. We have the following chain of equalities:

\[
\begin{aligned}
& \frac{1}{(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)\left(1+x^{8}\right)}=\frac{1-x}{1-x^{16}}= \\
& =(1-x)\left(1+x^{16}+x^{32}+x^{48}+\ldots\right)= \\
& =1-x+x^{16}-x^{17}+x^{32}-x^{33}+\ldots
\end{aligned}
\]

[M. S. Klamkin, M. M., 29, 53 (September 1955).]