Growth of recurrences with mixed multifold convolutions

Generalizing some popular sequences like Catalan’s number, Schröder’s number, etc, we consider the sequence $s_n$ with $s_0=1$ and for $n\ge 1$, \begin{multline*} s_n=\sum_{x_1+\dots+x_{\ell_1}=n-1} κ_1 s_{x_1}\dots s_{x_{\ell_1}} + \dots +\sum_{x_1+\dots+x_{\ell_{t’}}=n-1} κ_{t’} s_{x_1}\dots s_{x_{\ell_{t’}}}+\ \max_{x_1+\dots+x_{\ell_{t’+1}}=n-1} κ_{t’+1} s_{x_1}\dots s_{x_{\ell_{t’+1}}} + \dots + \max_{x_1+\dots+x_{\ell_t}=n-1} κ_t s_{x_1}\dots s_{x_{\ell_t}}, \end{multline*} where $x_i$ are nonnegative integers, ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_t$ are positive integers, and ${\kappa }_1,\dots ,{\kappa }_t$ are positive reals. We show that it is possible to compute the growth rate $\lambda$ of $s_n$ to any precision. In particular, for every $n\ge 2$, $$\sqrt[n]{\frac{{\kappa }^{\ast }}{\mathcal{L}(n-1)s_1}{s}_n}\le \lambda \le \sqrt[n]{3^{18\log 3+2\log \frac{s_1{\mathcal{L}}^2}{{\kappa }^{\ast }}}{n}^{3\log n+12\log 3+\log \frac{s_1{\mathcal{L}}^2}{{\kappa }^{\ast }}}{s}_n},$$where $\mathcal{L}={\max }_i{\mathrm{\ell }}_i$ and ${\kappa }^{\ast }={\kappa }_i$ for some $i$ with ${\mathrm{\ell }}_i\ge 2$, and the logarithm has the base $\frac{\mathcal{L}+1}{\mathcal{L}}$. The constants in the inequalities are not very well optimized and serve mostly as a proof of concept with the ratio of the upper bound and the lower bound converging to $1$ as $n$ goes to infinity.