A007004 a(n) = (3*n)! / ((n+1)*(n!)^3). (Formerly M3125)

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

Offset

0,2

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

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.

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

Example

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

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 *)

Prog

(Magma) [Factorial(3*n) / ((n+1)*Factorial(n)^3): n in [0..30]]; // Vincenzo Librandi, May 26 2011
(Maxima) makelist(multinomial_coeff(n, n, n)/(n+1), n, 0, 24); /* Emanuele Munarini, Oct 25 2016 */

Crossrefs

Cf. A000108, A060693.

Row n=3 of A215561.

Sequence in context: A212425 A336538 A294240 * A354291 A354287 A276361

Adjacent sequences: A007001 A007002 A007003 * A007005 A007006 A007007

Keyword

nonn

Author

N. J. A. Sloane, James Propp

Extensions

More terms from Zerinvary Lajos, May 27 2006

Status

approved