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.



Symmetric power series over $\mathbb{F}_2$
Let $I=\{i\_0\} \cup J$ and $g(x)=1+x+x^2+x^4+\dots$.


We show that adding the relation $\sum\_{i\in I}x\_i=0$, i.e. $x\_{i\_0}=\sum\_{i\ne i\_0}x\_i$ leads to an even degree power series: 


$$ f=g(x\_{i\_0})\prod\_{j\in J} g(x\_j) =\left(1+\sum\_{i\in J} (g(x\_i)-1)\right)\prod\_{j\in J} g(x\_j)\\ = \prod\_{j\in J} g(x\_j) + \sum\_{i\in J}\left(g(x\_i)-1\right)g(x\_i)\prod\_{j\in J\setminus \{i\}}g(x\_j)\\ = \prod\_{j\in J} g(x\_j) + \sum\_{i\in J}x\_i\prod\_{j\in J\setminus \{i\}}g(x\_j)= \prod\_{j\in J} g(x\_j) + \sum\_{i\in J}x\_i\frac{\partial}{\partial x\_i}\prod\_{j\in J}g(x\_j)$$
Now this is an even degree power series: check that each monomial with an odd number of $x\_j^1$ cancels.


