Consider the symmetric power series $$f = \prod_{i \in I}\left(1+x_i+x_i^2+x_i^4+x_i^8 + x_i^{16} +\ldots \right)$$ in variables $(x_i)_{i \in I}$ over $\mathbb F_2$. Fix some degree $r$, smaller than the number of variables, and denote the degree $r$ part of $p$ by $p^{(r)}$.

From looking at small-dimensional examples, I got the impression that the following might be true: If $r$ is odd, then $p^{(r)}$ is divisible by $\sigma_1 = \sum_{i \in I} x_i$. Could somebody provide a proof or counterexample of that statement? It would be interesting to me knowing the answer, either way.

Consider the symmetric power series $$f = \prod_{i \in I}\left(1+x_i+x_i^2+x_i^4+x_i^8 + x_i^{16} +\ldots \right)$$ in variables $(x_i)_{i \in I}$ over $\mathbb F_2$. Fix some degree $r$, smaller than the number of variables, and denote the degree $r$ part of $f$ by $f^{(r)}$.

From looking at small-dimensional examples, I got the impression that the following might be true: If $r$ is odd, then $f^{(r)}$ is divisible by $\sigma_1 = \sum_{i \in I} x_i$. Could somebody provide a proof or counterexample of that statement? It would be interesting to me knowing the answer, either way.