Bostan Mori

\[\begin{aligned} &[x^n] \frac{P(x)}{Q(x)} \\ \\ =&[x^n] \frac{P(x)Q(-x)}{Q(x)Q(-x)} \\ \\ =&[x^n] \frac{F_0(x^2) + xF_1(x^2)}{G(x^2)} \\ \\ =& \begin{cases} [x^{\lfloor n/2 \rfloor}] \frac{F_0(x)}{G(x)} & n \text{ is even} \\ \\ [x^{\lfloor n/2 \rfloor}] \frac{F_1(x)}{G(x)} & n \text{ is odd} \end{cases} \end{aligned} \]