# Second order recurrence relation for third order polynomial root

Consider this recurrence relation:

$$\begin{eqnarray*} f_0&=&1\\ f_n&=& \sum_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \ \text{for 1\leq n.} \end{eqnarray*}$$ where the Pochhammer symbol denotes the rising factorial. The generating function $f(z)=\sum_{n=0}^\infty f_nz^n$ seems to be a root of $$0=12 f^3 z^2- (f-1)^2 (f+2)$$ I have checked this to be true for the first 600 terms. However, I have been unable to come up with a proof. Do you have any ideas on how I might show this to be true?

Cheers, Petter

This is sequence A244038 in OEIS after scaling by $3^n$, so $f_n=(4/3)^n\binom{3n/2}n$. The fact that it satisfies a cubic equation is certainly a well-known result in hypergeometric functions.
EDIT: remove the "well-known": set $F(x)=\sum_{n\ge0}\binom{3n/2}{n}x^n$. Then $$F(x)=_2F_1(1/3,2/3;1/2;27x^2/4)+(3x/2)_2F_1(5/6,7/6;3/2,27x^2/4)$$ (which can probably be slightly simplified using contiguity relations), but can a hypergeometric expert explain why $(27x^2/4-1)F^3+3F-2=0$ ?