I am struggling to find the Taylor series at (0,0) of the function:
\begin{equation} \frac{1}{1+y-yx^2-2x^2y^2} \end{equation}
The first terms of the series are:
\begin{equation} \begin{array}{c} 1 + (x^2 - 1) y + (x^4 + 1) y^2 + (x^2 - 1) (x^2 + 1)^2 y^3 + \\ \\ + (x^8 + 2 x^6 - 2 x^4 + 2 x^2 + 1) y^4 + (x^{10} + 3 x^8 - 2 x^6 + 2 x^4 - 3 x^2 - 1) y^5 + O(y^6) \end{array} \end{equation}
so the resulting series should have the form:
\begin{equation} \sum_{n=0}^\infty P_n(x^2) y^n \end{equation}
where $P_n(x^2)$ is a polynomial expression. I would like to know the general form of this polynomial.