Write out the following equation as a power series

I am trying to write the following equation out as a power series:

$\frac{1}{x^{2}-3xn+3n^2}$

where $x\in \mathbb{R}$ and $n\in \mathbb{N}$ such that $x,n>0$. I have noticed that for $n=1$ we have

$\frac{1}{x^{2}-3xn+3n^2}=3+9x+24x^2+63x^3+...$

would there be a generalized formula for this expression as a power series in terms of $x$ and $n$?

Following @GEdgar's suggestion and trying to combine everything to make it as compact as possible $$\frac{1}{x^{2}-3xn+3n^2}=\sum_{p=0}^\infty a_p\,x^p$$ where $$a_p=\frac{2^{p-1} }{3 \,n^{2 (p+1)}}\Bigg[\left(1+ i\sqrt{3}\right) \left(\frac{3-i \sqrt{3}}{12} n\right)^p+\left(1- i\sqrt{3}\right)\left(\frac{3+i \sqrt{3}}{12} n\right)^p \Bigg]$$ For $n=1$, the coefficients would be $$\left\{\frac{1}{3},\frac{1}{3},\frac{2}{9},\frac{1}{9},\frac{1}{27},0,-\frac{1}{81} ,-\frac{1}{81},-\frac{2}{243},-\frac{1}{243},-\frac{1}{729},\cdots\right\}$$