A362170 a(n) = the hypergraph Catalan number C_5(n).

1, 1, 252, 299880, 1112865264, 11126161436292, 255654847841227632, 11676346013544951854304, 953196481551725431240711680, 128864126679853773803689954958112, 27235509875891350493949247236459319296, 8599544533810439129313490410035564948257536

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 = 5.

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=5 of A369288.

Cf. A000055, A000108, A362167, A362168, A362169, A362171, A362172.

Formula

a(n) ~ sqrt(5)/2 * (5^4/24)^n * n!^4/(Pi*n)^2 (conjectural).

Extensions

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