Is there a easy way to compute the coefficients of the power series which represents
\begin{equation*} \frac{x - x^k + x^{k+1}}{1-2x + x^k - x^{k+1}}. \end{equation*}
I am currently solving this by assuming it has the form
\begin{equation*} a_1x+a_2x^2+a_3x^3+\dots+a_nx^n+\dots \end{equation*}
and solving tedious recurrences obtained from
\begin{equation*} x - x^k + x^{k+1} = (1-2x + x^k - x^{k+1}) \times (a_1x+a_2x^2+a_3x^3+\dots+a_nx^n+\dots). \end{equation*}
Motivation: If my calculations are right, the above is the generating function of the number of compositions of an integer which do not involve a given integer $k$.