Generalizing some popular sequences like Catalan's number, Schr\"oder'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} \kappa_1 s_{ x_1}\dots s_{ x_{ \ell_1}} + \dots +\sum_{ x_1+\dots+x_{ \ell_{ t'}}=n-1} \kappa_{ t'} s_{ x_1}\dots s_{ x_{ \ell_{ t'}}}+\\ \max_{ x_1+\dots+x_{ \ell_{ t'+1}}=n-1} \kappa_{ t'+1} s_{ x_1}\dots s_{ x_{ \ell_{ t'+1}}} + \dots + \max_{ x_1+\dots+x_{ \ell_t}=n-1} \kappa_t s_{ x_1}\dots s_{ x_{ \ell_t}}, \end{ multline*} where $x_i$ are nonnegative integers, $\ell_1,\dots,\ell_t$ 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 $n\ge 2$, \[ \sqrt[n]{ \frac{ \kappa^*}{ \mathcal L(n-1) s_1} s_n} \le \lambda \le \sqrt[n]{ 3^{ 18\log 3 + 2\log\frac{ s_1\mathcal L^2}{ \kappa^*}} n^{ 3\log n + 12\log 3 + \log\frac{ s_1\mathcal L^2}{ \kappa^*}} s_n}, \]where $\mathcal L=\max_i \ell_i$ and $\kappa^*=\kappa_i$ for some $i$ with $\ell_i\ge 2$, 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.
Categories: math.CO