<s>[INST] Represent $f(x)=\frac{1}{(1-x^2)^4}$ as a power series Represent $f(x)=\frac{1}{(1-x^2)^4}$ as a power series
$$\sum\limits_{n=0}^{\infty}x^{2n}=\frac{1}{1-x^2}$$
Second derivative is
$$\left(\frac{1}{1-x^2}\right)^{''}=\frac{1}{(1-x^2)^4}\cdot x(1+8x-2x^2-8x^3+x^4)$$
This gives
$$\frac{1}{(1-x^2)^4}=\sum\limits_{n=0}^{\infty}\frac{2n(2n-1)x^{2n-2}}{x(1+8x-2x^2-8x^3+x^4)}$$
I have a proof in combinatorics which involves this series.
How to represent this series using binomial coefficients?
 [/INST] Perhaps it could be easier expand $(1-x^2)^{-4}$ using the binomial series.
$\begin{align}
(1-x^2)^{-4} &= 1+4x^2 + \frac{(-4)(-4-1)}{2}(-x^2)^2+\frac{(-4)(-4-1)(-4-2)}{3!}(-x^2)^3+ \dots\\
&=1+4x^2+10x^4+20x^6+35x^8+\dots\\
&=\sum_{n=1}^{\infty} \binom{n+2}{3}x^{2n-2}\\
\end{align}$
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