%I #17 Feb 01 2024 16:37:22

%S 1,1,20,860,57200,5344800,682612800,118180104000,27396820448000,

%T 8312583863720000,3209035788149600000,1534218535286625760000,

%U 888028389273314675200000,611029957551257895664000000,492466785518772137553984000000,459270692175324078697443840000000

%N a(n) = the hypergraph Catalan number C_3(n).

%C 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 = 3.

%C Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.

%H Andrew Howroyd, <a href="/A362168/b362168.txt">Table of n, a(n) for n = 0..200</a>

%H Paul E. Gunnells, <a href="https://arxiv.org/abs/2102.05121">Generalized Catalan numbers from hypergraphs</a>, arXiv:2102.05121 [math.CO], 2021.

%F a(n) ~ sqrt(3) * (9/2)^n * n!^2/(Pi*n) (conjectural).

%o (PARI) Vec(HypCatColGf(3,15)) \\ HypCatColGf defined in A369288. - _Andrew Howroyd_, Feb 01 2024

%Y Column k=3 of A369288.

%Y Cf. A000055, A000108, A362167, A362169, A362170, A362171, A362172.

%K nonn,walk

%O 0,3

%A _Peter Bala_, Apr 10 2023

%E a(9) onwards from _Andrew Howroyd_, Feb 01 2024