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A362168
a(n) = the hypergraph Catalan number C_3(n).
7
1, 1, 20, 860, 57200, 5344800, 682612800, 118180104000, 27396820448000, 8312583863720000, 3209035788149600000, 1534218535286625760000, 888028389273314675200000, 611029957551257895664000000, 492466785518772137553984000000, 459270692175324078697443840000000
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 = 3.
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
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.
FORMULA
a(n) ~ sqrt(3) * (9/2)^n * n!^2/(Pi*n) (conjectural).
PROG
(PARI) Vec(HypCatColGf(3, 15)) \\ HypCatColGf defined in A369288. - Andrew Howroyd, Feb 01 2024
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Peter Bala, Apr 10 2023
EXTENSIONS
a(9) onwards from Andrew Howroyd, Feb 01 2024
STATUS
approved