Expand rational fractional expression in power series $\frac{k^2}{(k^2+\frac{1}{2})^{n+2}}$

Question

Expression as showed in the title and n can be take arbitrary natural numbers, 0,1,2,.... How to expand it in powers of k and what coefficient of $k^m$ is? Any suggestion is much appreciated.

Answer 1

$$\text{coefficient of $x^m$ in }\frac{k^2}{\left(k^2+\frac{1}{2}\right)^{n+2}}=\text{coefficient of $x^{m-2}$ in }\left(k^2+\frac{1}{2}\right)^{-(n+2)}=\begin{cases}2^{n+2+(m-2)/2}\;^{-(n+2)}{\rm C}_{(m-2)/2}&\text{m%2==0}\\0&\text{m%2!=0}\end{cases}$$ Because $$(1+x)^{-n}=\sum_{\nu=0}^{\infty}\;^{-n}{\rm C}_{\nu}x^{\nu}$$