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

Author

Clark Kimberling

Keywords

• nonn

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.

Programs

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 *)

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

Extensions

More terms from Alois P. Heinz, Feb 20 2012