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%I M3125 #72 Sep 08 2022 08:44:35
%S 1,3,30,420,6930,126126,2450448,49884120,1051723530,22787343150,
%T 504636071940,11377249621920,260363981732400,6034149862347600,
%U 141371511060715200,3343436236585914480,79726203788589122490,1914992149823954412750,46295775130831740013500
%N a(n) = (3*n)! / ((n+1)*(n!)^3).
%C Number of walks with steps (0,1)/North, (1,0)/East and (-1,-1)/Southwest from (0,0) to (0,0) of length 3n, and staying above the line y=x (i.e., any point (x,y) along the walk satisfies y>=x ). - _Shanzhen Gao_, Nov 09 2010
%C Number of walks in 3-dimensions using steps (1,0,0), (0,1,0), and (0,0,1) from (0,0,0) to (n,n,n) such that after each step we have y<=x. - _Eric Werley_, Jun 24 2011
%C Number of possible necklaces consisting of n white beads, n-1 red beads and n-1 black beads, where two necklaces are considered equivalent if they differ by a cyclic permutation. - _Thotsaporn Thanatipanonda_, Feb 20 2011
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A007004/b007004.txt">Table of n, a(n) for n = 0..150</a>
%H Alin Bostan, <a href="https://specfun.inria.fr/bostan/HDR.pdf">Calcul Formel pour la Combinatoire des Marches</a> [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017.
%F a(n) = C(2*n,n)*C(3*n,n)/(n+1) = A000108(n)*C(3*n,n). - _Zerinvary Lajos_, May 27 2006
%F a(n) = A060693(2n,n) = A088617(2n,n). - _Philippe Deléham_, Nov 23 2011
%F a(n) = (3*(3*n-1)*(3*n-2)*a(n-1))/(n*(n+1)) for n>0, a(0)=1. - _Alois P. Heinz_, Aug 13 2013
%F a(n) ~ 3^(3*n+1/2)/(2*Pi*n^2). - _Vaclav Kotesovec_, Sep 06 2016
%e n=1, three walks: NE(SW), (SW)NE, N(SW)E. - _Shanzhen Gao_, Nov 09 2010
%p seq(binomial(2*n,n)*binomial(3*n,n)/(n+1), n=0..20); # _Zerinvary Lajos_, May 27 2006
%t a[n_]:=(3*n)!/((n + 1)*(n!)^3); (* _Vladimir Joseph Stephan Orlovsky_, Dec 13 2008 *)
%t CoefficientList[Series[Hypergeometric2F1[1/3,2/3,2,27 x],{x,0,20}],x] (* _Harvey P. Dale_, Apr 07 2013 *)
%t Table[Multinomial[n, n, n]/(n + 1), {n, 0, 12}] (* _Emanuele Munarini_, Oct 25 2016 *)
%o (Magma) [Factorial(3*n) / ((n+1)*Factorial(n)^3): n in [0..30]]; // _Vincenzo Librandi_, May 26 2011
%o (Maxima) makelist(multinomial_coeff(n,n,n)/(n+1),n,0,24); /* _Emanuele Munarini_, Oct 25 2016 */
%Y Cf. A000108, A060693.
%Y Row n=3 of A215561.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, _James Propp_
%E More terms from _Zerinvary Lajos_, May 27 2006
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