A024489 a(n) = (1/(9n-3))*M(3n; n,n,n), where M() is a multinomial coefficient.

1, 6, 70, 1050, 18018, 336336, 6651216, 137181330, 2921454250, 63804560820, 1422156202740, 32235540595440, 741035948007600, 17240428178136000, 405264998374050240, 9612379180184504130, 229799057978874529530, 5532199543935868303500, 134014085905039247407500

Offset

1,2

Comments

a(n) is also the number of possible necklaces consisting of n white beads, n red beads and n-1 black beads, where two necklaces are considered equivalent if they differ by a cyclic permutation. - Thotsaporn Thanatipanonda, Feb 20 2012

Links

Alois P. Heinz, Table of n, a(n) for n = 1..300

Yang-Hui He, Vishnu Jejjala, Cyril Matti, Brent D. Nelson, Michael Stillman, The geometry of generations, Commun Math. Phys. 339 (1) (2015) 149-190

Paul Tarau, Valeria de Paiva, Deriving Theorems in Implicational Linear Logic, Declaratively, arXiv:2009.10241 [cs.LO], 2020.

Formula

a(n) ~ 3^(3*n-3/2) / (2*Pi*n^2). - Vaclav Kotesovec, Aug 25 2014

a(n) = (3*n)!/(n!^3*(9*n-3)). - Peter Luschny, Sep 30 2018

D-finite with recurrence n^2*a(n) -3*(3*n-2)*(3*n-4)*a(n-1)=0. - R. J. Mathar, Jan 14 2021

Maple

with(combinat):
a:= n-> multinomial(3*n, n$3)/(9*n-3):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 20 2012

Mathematica

a[n_] := (3n)!/((9n-3) n!^3); Array[a, 20] (* Jean-François Alcover, Jun 01 2019 *)

Crossrefs

Sequence in context: A104900 A186667 A001448 * A354328 A036361 A365057

Adjacent sequences: A024486 A024487 A024488 * A024490 A024491 A024492

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Alois P. Heinz, Feb 20 2012

Status

approved