Consider the following functional equation where $n \in \mathbb{N}$ is fixed,
$$ F_n(x^{-1},xy) = \frac{xy-y}{(1-xy+y)(1+y)^n} + \frac{1}{(1-xy+y)^{n+1}} F\left (x,\frac{y}{1-xy+y} \right ). $$
I know that $$ F_1(x,y) = \frac{2}{y} \cdot \frac{(2xy+1)y + \sqrt{1-2(2xy+1)y + y^2} - 1}{1- (2xy+1)^2} $$ is the solution for $n=1$.
(this was suggested to me by OEIS since the recursion happens to correspond to a known combinatorial problem).
Is it possible to find an explicit solution for general $n$?