Closed form expression for series $f_\alpha(x) = \sum_{n\geq 0} (-n)^{\alpha} x^{2n}$

Let $\alpha \in \mathbb N$, and consider the Taylor series$$f_\alpha(x) = \sum_{n >0} (-n)^{\alpha} x^{2n}$$which is convergent for $|x|<1$.

Question: can we find a closed form to express $f_\alpha(x)$?

I worked out two cases:$$\alpha = 0 \Rightarrow f_\alpha(x) = \frac{x^2}{(1-x^2)}$$ andif $$\alpha = 1 \Rightarrow f_\alpha(x) = \frac{-x^2}{(1-x^2)^2}$$

I don't see however how to get to the general $\alpha$.