A362172 a(n) = the hypergraph Catalan number C_7(n).

1, 1, 3432, 141858288, 40309820014464, 53321581727982247680, 238681094467043912358445056, 2924960829706245011243295851200512, 84750120431280677998861681616641721991168, 5208807724759446156144077076658272647436908396544

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 = 7.

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..80
• Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

Crossrefs

Column k=7 of A369288.

Cf. A000055, A000108, A362167, A362168, A362169, A362170, A362171.

Formula

a(n) ~ sqrt(7)/4 * (7^6/6!)^n * n!^6/(Pi*n)^3 (conjectural).

Extensions

a(7) onwards from Andrew Howroyd, Feb 01 2024