Consider this recurrence relation:
$$ \begin{eqnarray*} f_1&=&1\\ f_n&=& \sum_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \ \text{for $1\leq n$.} \end{eqnarray*} $$$$ \begin{eqnarray*} f_0&=&1\\ f_n&=& \sum_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \ \text{for $1\leq n$.} \end{eqnarray*} $$ 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
