cp's OEIS Frontend

A362167 a(n) = the hypergraph Catalan number C_2(n).

%I A362167 #30 May 07 2024 07:05:42
%S A362167 1,1,6,57,678,9270,139968,2285073,39871926,739129374,14521778820,
%T A362167 302421450474,6687874784484,157491909678168,3961138908376692,
%U A362167 106663881061254465,3078671632202791782,95213375569840078422,3149291101933230285924,111073721303120881912686
%N A362167 a(n) = the hypergraph Catalan number C_2(n).
%C A362167 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 A362167 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 A362167 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 A362167 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 A362167 Andrew Howroyd, <a href="/A362167/b362167.txt">Table of n, a(n) for n = 0..200</a>
%H A362167 Paul E. Gunnells, <a href="https://arxiv.org/abs/2102.05121">Generalized Catalan numbers from hypergraphs</a>, arXiv:2102.05121 [math.CO], 2021.
%F A362167 a(n) ~ e^(3/2) * 2^(n+1) * n!/sqrt(Pi*n) (conjectural).
%o A362167 (PARI) Vec(HypCatColGf(2, 20)) \\ HypCatColGf defined in A369288. - _Andrew Howroyd_, Feb 06 2024
%Y A362167 Column k=2 of A369288.
%Y A362167 Cf. A000055, A000108, A362168, A362169, A362170, A362171, A362172.
%K A362167 nonn,walk
%O A362167 0,3
%A A362167 _Peter Bala_, Apr 10 2023
%E A362167 a(11) onwards from _Andrew Howroyd_, Jan 31 2024