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 = 4.
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(2) * (32/3)^n * n!^3/(Pi*n)^(3/2) (conjectural).
PROG
(PARI) Vec(HypCatColGf(4, 15)) \\ HypCatColGf defined in A369288. - Andrew Howroyd, Feb 01 2024
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Peter Bala, Apr 10 2023
EXTENSIONS
a(8) onwards from Andrew Howroyd, Feb 01 2024
STATUS
approved