Similar to questions asked here and here, I'm trying to solve a quadratic recurrence relation that looks similar to the Catalan recurrence:
$$ C_n = \sum_{p=0}^{n-1} \alpha_p \ C_p \ C_{n-1-p} $$
where the coefficients $\alpha_n$ are some known sequence of rational numbers. I can solve this for a few simple cases (including the obvious case $\alpha_n$ = 1, and the fairly straightforward cases $\alpha_n = n$ and $\alpha_n = \frac{1}{n+1}$), but I'm wondering if there are closed-form solutions for more general choices of $\alpha_n$.
For tractibility, let's assume the $\alpha_n$ are monotonic (increasing or decreasing) with $n$ (otherwise the solution is probably chaotic or something).
For some context, this is coming from a series solution to a nonlinear integral equation of the form:
$$ 1 = G \ f(x) + G \ \mathcal{L}[G] $$
where $G(x)$ is the unknown function, $f(x)$ is a known function, and $\mathcal{L}$ is a linear integral operator. The coefficients $C_n$ are related to the coefficients of the series expansion:
$$ G(x) = \sum_n c_n x^n $$