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) = (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
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Zerinvary Lajos, May 27 2006
STATUS
approved