Generating function and integer sequence that arise from this function

Question

I am looking for the power series arising from the generating function $f(x)$ that solves the following equation:

$\alpha^{2}x^{3}+3\alpha x^{2}f(x)+3xf(x)^{2}=\beta$

for some $\alpha,\beta\in \mathbb{R}$. What would be some possibilities for integers sequences that arise from the generating function of the above equation? I understand that when solving for $f(x)$ one gets

$f(x)=\frac{-3\alpha x^2\pm \sqrt{-3\alpha^2x^4+12x\beta}}{6x}$

but I am wondering what would be the associated integer sequence. The power series computation here seems to be quite tedious. I want to write $f(x)$ as

$f(x)=\sum_{k\ge 0}a_kx^k$

where $a_k$ for each $k$ is the $k$th member of the associated sequence.

Answer 1

The series for $$ f(x)=\frac{-3\alpha x^2\pm \sqrt{-3\alpha^2x^4+12x\beta}}{6x} $$ near $x=0$ looks like $$ f(x) = \pm\frac{\sqrt\beta}{\sqrt3\sqrt x} - \frac{\alpha x}{2} \mp \frac{\alpha^2 x^{5/2}}{8\sqrt3\sqrt\beta}+\dots $$ which does not seem to be the type of series you want?

Answer 2

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 $$\frac{-3\alpha\pm\sqrt{3}i|\alpha|}{6}$$