On the Asymptotics of Takeuchi Numbers

Abstract

I present an asymptotic formula for the Takeuchi numbers $T_n$. In particular, I give compelling numerical evidence and present a heuristic argument showing that $$T_n\sim C_T{B}_n\exp \frac{1}{2}{W(n)}^2$$as $n$ tends to infinity, where $B_n$ are the Bell numbers, W(n) is Lambert's $W$ function, and $C_T=\mathrm{2.239...}$ is a constant. Moreover, I show that the method presented here can be generalized to derive conjectures for related problems.

latex file with 13 pages, 2 postscript figures included