Growth of recurrences with mixed multifold convolutions
Vuong Bui
Abstract
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, $\ell_1,\dots,\ell_t$ are positive integers, and $κ_1,\dots,κ_t$ are positive reals. We show that it is possible to compute the growth rate $λ$ of $s_n$ to any precision. In particular, for every $n\ge 2$, \[ \sqrt[n]{\frac{κ^*}{\mathcal L(n-1) s_1} s_n} \le λ\le \sqrt[n]{3^{18\log 3 + 2\log\frac{s_1\mathcal L^2}{κ^*}} n^{3\log n + 12\log 3 + \log\frac{s_1\mathcal L^2}{κ^*}} s_n}, \]where $\mathcal L=\max_i \ell_i$ and $κ^*=κ_i$ 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.