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a(n) = the hypergraph Catalan number C_2(n).
11

%I #30 May 07 2024 07:05:42

%S 1,1,6,57,678,9270,139968,2285073,39871926,739129374,14521778820,

%T 302421450474,6687874784484,157491909678168,3961138908376692,

%U 106663881061254465,3078671632202791782,95213375569840078422,3149291101933230285924,111073721303120881912686

%N a(n) = the hypergraph Catalan number C_2(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. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 2.

%C Let T(n) be the set of unlabeled trees on n vertices (see A000055). Let T be a tree in T(n+1), and let v be a vertex of T. Then an a(m,T)-tour beginning at v is a walk that begins and ends at v and traverses each edge of T exactly 2*m times. We denote by a(m,T)(v) the number of a(m,T)-tours beginning at v.

%C The hypergraph Catalan numbers C_m(n) are defined by C_m(n) = Sum_{trees T in T(n+1)} Sum_{vertices v in T} a(m,T)(v)/|Aut(T)|, where Aut(T) denotes the automorphism group of the tree T.

%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="/A362167/b362167.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) ~ e^(3/2) * 2^(n+1) * n!/sqrt(Pi*n) (conjectural).

%o (PARI) Vec(HypCatColGf(2, 20)) \\ HypCatColGf defined in A369288. - _Andrew Howroyd_, Feb 06 2024

%Y Column k=2 of A369288.

%Y Cf. A000055, A000108, A362168, A362169, A362170, A362171, A362172.

%K nonn,walk

%O 0,3

%A _Peter Bala_, Apr 10 2023

%E a(11) onwards from _Andrew Howroyd_, Jan 31 2024