Your generating function is incorrect.
You have the recurrence $a_n=a_{n-2}-2a_{n-3}+2[n=2]\;,$ where the last term is an Iverson bracket, and I assume that $a_n=0$ for $n<0$. Then
$\begin{align*} A(x)&=\sum_{n\ge 0}a_nx^n\\ &=\sum_{n\ge 0}a_{n-2}x^n-2\sum_{n\ge 0}a_{n-3}x^n+2x^2\\ &=x^2A(x)-2x^3A(x)+2x^2\;, \end{align*}$
so $A(x)=\frac{2x^2}{1-x^2+2x^3}\;.$
Your function is
$\begin{align*} \frac{-2x^4}{1+x^2}&=-2x^4\cdot\frac1{1-(-x^2)}\\ &=-2x^4\sum_{n\ge 0}(-x^2)^n\\ &=-2x^4\sum_{n\ge 0}(-1)^nx^{2n}\;, \end{align*}$
in which the coefficient of $x^2$ and of every odd power of $x$ is $0$. However, $a_2=2\ne 0$, and $a_3=a_1-2a_0=0$, so $a_5=a_3-2a_2=-4\ne 0$.