Consider this recurrence relation:

$$
\begin{eqnarray*}
f_1&=&1\\
 f_n&=&
\sum_{m=0}^{n-1}  \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \  \text{for $1\leq n$.}
\end{eqnarray*}
$$
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