A362168 a(n) = the hypergraph Catalan number C_3(n).

1, 1, 20, 860, 57200, 5344800, 682612800, 118180104000, 27396820448000, 8312583863720000, 3209035788149600000, 1534218535286625760000, 888028389273314675200000, 611029957551257895664000000, 492466785518772137553984000000, 459270692175324078697443840000000

Offset: 0

Author

Peter Bala, Apr 10 2023

Keywords

• nonn
• walk

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 = 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.

Links

• 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.

Crossrefs

Column k=3 of A369288.

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

Programs

PARI

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

Formula

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

Extensions

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