A functional power series equation

A functional power series equation

I am interested in solving the following functional equation:
$$(1-z^2-wz^2)F(z,w)=w-wz^2-wz^{2n}F(z^2,w)\,.$$
Here $n$ is an integer with $n\geq2$ and $F(z,w)$ is a power series in two
variables $z,w$. I am very tired to try to solve the infinite system of
equations satisfied by the coefficients of $F$ (tough it seems that this
is in fact a linear system, am I wrong?).
Any solution is welcomed, using complex variable methods, for example
(that is, thinking $F$ as a holomorphic function in two variables defined
in some neighborhood of the origin in $\mathbb C^2$). I am also interested
in the case $n=1$, but surely the hard work involved in the case $n\geq2$
can be imitated in order to solve it.
EDIT
One approach is to determine the values of the following infinite product
and the following series:
$$\lim_{r\to\infty}A_r,\ \text{where}\
A_r=\prod_{k=0}^r\frac{-wz^{2^{k+1}n}}{1-z^{2^k}-wz^{2^{k+1}}}\,,$$
and
$$\sum_{r=0}^\infty B_r,\ \text{where}\
B_r=A_r\,\frac{w-wz^{2^{r+1}}}{1-z^{2^r}-wz^{2^{r+1}}}\,.$$
In fact, defining $G(j)=F(z^j,w)$ and after some algebra (which I am very
lazy to typing) I obtain the following equality:
$$F(z,w)=G(1)=\lim_{r\to\infty}A_rF(z^{2^r},w)+\sum_{r=0}^\infty B_r\,.$$
The functional equation implies that $F(0,w)=w$, and since $F$ is
holomorphic (believe me!), then we have $w=\lim_{r\to\infty}F(z^{2^r},w)$
for $|z|$ small, so if $A_r$ converges to $A(z,w)$ and $\sum_{r=0}^\infty
B_r=B(z,w)$, then $F(z,w)=wA(z,w)+B(z,w)$, so it remains to determine both
$A(z,w)$ and $B(z,w)$ (and pray that $A$ and $B$ have nice formulas).