A362171 a(n) = the hypergraph Catalan number C_6(n).

1, 1, 924, 6358044, 203356067376, 23345633108619360, 7484535614458774428480, 5583028528736289502562408256, 8547031978688473343843434600852224, 24503310825110075324451531207978424853568, 122607946140627185219752569884701085604290069760

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 = 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.

Crossrefs

Column k=6 of A369288.

Cf. A000055, A000108, A362167, A362168, A362169, A362170, A362172.

Formula

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

Extensions

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