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A362169
a(n) = the hypergraph Catalan number C_4(n).
7
1, 1, 70, 15225, 7043750, 6327749750, 10411817136000, 29034031694460625, 126890003304310093750, 816448082514611102718750, 7379204202189710013311562500, 90369225128606332243844280406250, 1457163640851863433667228849319062500, 30217741884769257764596041337071409375000
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 = 4.
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(2) * (32/3)^n * n!^3/(Pi*n)^(3/2) (conjectural).
PROG
(PARI) Vec(HypCatColGf(4, 15)) \\ HypCatColGf defined in A369288. - Andrew Howroyd, Feb 01 2024
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Peter Bala, Apr 10 2023
EXTENSIONS
a(8) onwards from Andrew Howroyd, Feb 01 2024
STATUS
approved