Generalization of x^n + 1/x^n = k

I am not sure what kind of answer you are looking for

I expanded    $(x+\frac{1}{x})^n$

and got

$x^n+\frac{1}{x^n}=k^n-\displaystyle \sum_{r=1}^{n-1}\binom{n}{r} x^{n-2r}$