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

1, 1, 70, 15225, 7043750, 6327749750, 10411817136000, 29034031694460625

Offset

0,3

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

Table of n, a(n) for n=0..7.

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

Formula

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

Crossrefs

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

Sequence in context: A330244 A362916 A229776 * A231054 A007100 A103157

Adjacent sequences: A362166 A362167 A362168 * A362170 A362171 A362172

Keyword

nonn,walk,more

Author

Peter Bala, Apr 10 2023

Status

approved