A007004 a(n) = (3*n)! / ((n+1)*(n!)^3).

1, 3, 30, 420, 6930, 126126, 2450448, 49884120, 1051723530, 22787343150, 504636071940, 11377249621920, 260363981732400, 6034149862347600, 141371511060715200, 3343436236585914480, 79726203788589122490, 1914992149823954412750, 46295775130831740013500

Offset: 0

Author

N. J. A. Sloane, James Propp

Keywords

• nonn

Comments

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

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

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

Examples

			n=1, three walks: NE(SW), (SW)NE, N(SW)E. - _Shanzhen Gao_, Nov 09 2010
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..150
• Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d'Informatique de Paris Nord, Université Paris 13, December 2017.

Crossrefs

Cf. A000108, A060693.

Row n=3 of A215561.

Programs

Magma

[Factorial(3*n) / ((n+1)*Factorial(n)^3): n in [0..30]]; // Vincenzo Librandi, May 26 2011

Maple

seq(binomial(2*n,n)*binomial(3*n,n)/(n+1), n=0..20); # Zerinvary Lajos, May 27 2006

Mathematica

a[n_]:=(3*n)!/((n + 1)*(n!)^3); (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
CoefficientList[Series[Hypergeometric2F1[1/3,2/3,2,27 x],{x,0,20}],x] (* Harvey P. Dale, Apr 07 2013 *)
Table[Multinomial[n, n, n]/(n + 1), {n, 0, 12}] (* Emanuele Munarini, Oct 25 2016 *)

Maxima

makelist(multinomial_coeff(n,n,n)/(n+1),n,0,24); /* Emanuele Munarini, Oct 25 2016 */

Formula

a(n) = C(2*n,n)*C(3*n,n)/(n+1) = A000108(n)*C(3*n,n). - Zerinvary Lajos, May 27 2006

a(n) = A060693(2n,n) = A088617(2n,n). - Philippe Deléham, Nov 23 2011

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

a(n) ~ 3^(3*n+1/2)/(2*Pi*n^2). - Vaclav Kotesovec, Sep 06 2016

Extensions

More terms from Zerinvary Lajos, May 27 2006