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a(n) = (1/(9n-3))*M(3n; n,n,n), where M() is a multinomial coefficient.
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%I #30 Jan 14 2021 08:04:40

%S 1,6,70,1050,18018,336336,6651216,137181330,2921454250,63804560820,

%T 1422156202740,32235540595440,741035948007600,17240428178136000,

%U 405264998374050240,9612379180184504130,229799057978874529530,5532199543935868303500,134014085905039247407500

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

%C 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

%H Alois P. Heinz, <a href="/A024489/b024489.txt">Table of n, a(n) for n = 1..300</a>

%H Yang-Hui He, Vishnu Jejjala, Cyril Matti, Brent D. Nelson, Michael Stillman, <a href="https://doi.org/10.1007/s00220-015-2416-7">The geometry of generations</a>, Commun Math. Phys. 339 (1) (2015) 149-190

%H Paul Tarau, Valeria de Paiva, <a href="https://arxiv.org/abs/2009.10241">Deriving Theorems in Implicational Linear Logic, Declaratively</a>, arXiv:2009.10241 [cs.LO], 2020.

%F a(n) ~ 3^(3*n-3/2) / (2*Pi*n^2). - _Vaclav Kotesovec_, Aug 25 2014

%F a(n) = (3*n)!/(n!^3*(9*n-3)). - _Peter Luschny_, Sep 30 2018

%F 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

%p with(combinat):

%p a:= n-> multinomial(3*n, n$3)/(9*n-3):

%p seq(a(n), n=1..20); # _Alois P. Heinz_, Feb 20 2012

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

%K nonn

%O 1,2

%A _Clark Kimberling_

%E More terms from _Alois P. Heinz_, Feb 20 2012