Polynomials over $\mathbb F_2$ without zeros in $\mathbb F_2$ having an inverse series with support of large density. Does there exist a sequence $A_n=A_n(x)\in\mathbb F_2[x]$ over the field $\mathbb F_2$ of 
two elements (represented by $0$ and $1$) such that $A_n(0)=A_n(1)=1$ and the inverse series $1/A_n=\sum_{j=0}^\infty
\alpha_{n,j}x^j\in \mathbb F_2[[x]]$
have supports with densities
$\delta_n=\lim_{k\rightarrow\infty} \frac{\alpha_{n,0}+\alpha_{n,1}+\dots+\alpha_{n,k-1}}{k}$ 
converging to $1$?
A positive answer to this question would give a positive answer to question 
[Sum of densities of support of $A$ and $A^{-1}$ for $A=1+\dots\in \mathbb F_2[[x]]$  by considering $\frac{A_n(x^2)}{1+x}$.
The highest possible density for polynomials of degree $\leq 16$ is $\frac{2}{3}$,
achieved by $1+x+x^2$.
 A: This is not an answer, rather a possible suggestion on how to deal with irreducible polynomials $A(x)$: Let $A(x)\in\mathbb F_2[x]$ be irreducible of degree $n$. Then \begin{equation}
A(x)=\prod_{i=0}^{n-1}(1+\lambda^{2^i}x)
\end{equation}
for some $\lambda\in\mathbb F_{2^n}$. The partial fraction decomposition and geometric series yield
\begin{equation}
\frac{1}{A(x)}=\sum_{i=0}^{n-1}\frac{\alpha^{2^i}}{1+\lambda^{2^i}x}
= \sum_{m=0}^\infty\sum_{i=0}^{n-1}\alpha^{2^i}(\lambda^{2^i}x)^m
= \sum_{m=0}^\infty T(\alpha\lambda^m)x^m,
\end{equation}
where $\alpha=\lambda/f'(1/\lambda)$ and $T$ is the trace map from $\mathbb F_{2^n}$ to $\mathbb F_2$.
Note that the power series is periodic with period $e$, where $e$ is the multiplicative order of $\lambda$. Thus if $U$ is the subgroup of order $e$ of $\mathbb F_{2^n}^\star$, then the density of $1$'s is the number of $u\in U$ with $T(\alpha u)=1$ divided by $\lvert U\rvert$.
An easy case is when $e=2^n-1$, so $U=\mathbb F_{2^n}^\star$. Half of the elements of $\mathbb F_{2^n}$ have trace $0$, so the density of $1$'s is $2^{n-1}/(2^n-1)$.
So when not only $n$ is prime, but even $2^n-1$ is prime, then we have this case and the density is only slightly bigger than $1/2$.
The general case seems to be more challenging. It is always difficult to relate an additive function like the trace map with subgroups of the multiplicative group of fields.
