A362169 a(n) = the hypergraph Catalan number C_4(n).

1, 1, 70, 15225, 7043750, 6327749750, 10411817136000, 29034031694460625, 126890003304310093750, 816448082514611102718750, 7379204202189710013311562500, 90369225128606332243844280406250, 1457163640851863433667228849319062500, 30217741884769257764596041337071409375000

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 = 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.

Crossrefs

Column k=4 of A369288.

Cf. A000055, A000108, A362167, A362168, A362170, A362171, A362172.

Programs

PARI

Vec(HypCatColGf(4,15)) \\ HypCatColGf defined in A369288. - Andrew Howroyd, Feb 01 2024

Formula

a(n) ~ sqrt(2) * (32/3)^n * n!^3/(Pi*n)^(3/2) (conjectural).

Extensions

a(8) onwards from Andrew Howroyd, Feb 01 2024