# Generating function and integer sequence that arise from this function

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.

• Are you just trying to solve the given equation for $f(x)$? If so, it's just a quadratic equation.
– Karl
Nov 22, 2021 at 17:33
• I should have probably specified that in particular, I am looking for the integer sequence that arises from the generating function itself. I've edited my question.
– user931598
Nov 22, 2021 at 17:41

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