OFFSET

0,3

COMMENTS

Let m >= 1. The sequence of hypergraph Catalan numbers {C_m(n): n >= 0} is defined in terms of counting walks on trees, weighted by the orders of their automorphism groups. See Gunnells. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 6.

Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..100

Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

FORMULA

a(n) ~ sqrt(3)/2 * (6^5/5!)^n * n!^5/(Pi*n)^(5/2) (conjectural)

CROSSREFS

KEYWORD

nonn,walk

AUTHOR

Peter Bala, Apr 10 2023

EXTENSIONS

a(6) onwards from Andrew Howroyd, Feb 01 2024

STATUS

approved