Generating function and integer sequence that arise from this function
First note that for $\beta\neq0$ there is no formal power series $f$ satisfying the equation. This is because the left hand side belongs to the maximal ideal generated by $x$ while the right hand side is a unit.
Let's assume that $\beta=0$. In this case you get the solutions $$f(x)=\frac{-3\alpha\pm\sqrt{3}i|\alpha|}{6}x$$
Therefore, the corresponding sequences are all zero except for the $1$-th term which is one of the values $$\frac{-3\alpha\pm\sqrt{3}i|\alpha|}{6}$$