0,3

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 = 3.

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

Andrew Howroyd, Table of n, a(n) for n = 0..200

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

a(n) ~ sqrt(3) * (9/2)^n * n!^2/(Pi*n) (conjectural).

(PARI) Vec(HypCatColGf(3, 15)) \\ HypCatColGf defined in A369288. - Andrew Howroyd, Feb 01 2024

Column k=3 of A369288.

Cf. A000055, A000108, A362167, A362169, A362170, A362171, A362172.

Sequence in context: A117798 A006424 A291112 * A192370 A113102 A285673

Adjacent sequences: A362165 A362166 A362167 * A362169 A362170 A362171

nonn,walk

Peter Bala, Apr 10 2023

a(9) onwards from Andrew Howroyd, Feb 01 2024

approved