Hello guys :) I've got a bit of an issue with the following recurrence relation:
Let $\omega _1 = 1$. Now, let $\omega _n$ be defined as follows $$ w_n = 1 + \sum _{k=1}^{n-1}C^{k}_{n-k} \, \omega _k \, \omega _{n-k} $$ where $$ C^{k}_{n-k} = \frac{1+\delta^{k}_{n-k}}{2} = \left\lbrace \begin{array}{l l} 1/2, & n \neq n-k \\ 1, & n= n-k \end{array} \right. $$ My goal was to (at least attempt) to find a closed form for $\omega _n$, but it has proven quite difficult for me. I've tried rewriting the term $C^{k}_{n-k} \, \omega _k \, \omega _{n-k}$ as the application of a bilinear form, approximating the sum as a weighted convolution by setting $\omega _0 = 0$ artificially, and some other things but it just won't budge. To be honest, my repertoire of recurrence tricks isn't too vast, and I've read online that only a few classes of nonlinear recurrence relations actually have a closed form. Any help and kind observations would be appreciated :)
By looking at the first few terms and using the gfun package in Maple, it appears that the generating function of your sequence is
$$ g(x) = \sum_{n=1}^\infty \omega_n x^n = 1 - \sqrt{\frac{1-3x}{1-x}} $$