OFFSET
2,1
COMMENTS
Number of standard tableaux of shapes (n-1,n-1,k), k=0,1,...,n-1. Example: a(3)=12 because there are 2, 5 and 5 standard tableaux of shapes (2,2), (2,2,1) and (2,2,2), respectively. - Emeric Deutsch, May 25 2004
From Joel B. Lewis, Oct 05 2009: (Start)
Also the number of standard shifted Young tableaux of shape (n+1, n, n-1).
Also the number of 2143-avoiding up-down permutations of length 2n - 1. (End)
LINKS
J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.
F. Ruskey, Generating linear extensions of posets by transpositions, J. Combin. Theory, B 54 (1992), 77-101.
Dennis White, Sign-balanced posets
FORMULA
a(n) = multinomial(3n; n-1, n, n+1)/(n(2n-1)(2n+1)).
a(n) ~ 3^(3*n+1/2) / (8*Pi*n^4). - Vaclav Kotesovec, Nov 13 2014
D-finite with recurrence n*(n+1)*(2*n+1)*a(n) -3*(3*n-1)*(2*n-3)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Jul 07 2023
MAPLE
A007724 := proc(n)
combinat[multinomial](3*n, n-1, n, n+1)/n/(2*n-1)/(2*n+1) ;
end proc:
seq(A007724(n), n=2..40) ; # R. J. Mathar, Jul 07 2023
MATHEMATICA
Table[(3*n)!/((n-1)!*n!*(n+1)!)/(n*(2*n-1)*(2*n+1)), {n, 2, 10}] (* Vaclav Kotesovec, Nov 13 2014 *)
Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3}, 1], {n, 19}] (* Michael De Vlieger, Aug 22 2016 *)
PROG
(PARI) {a(n) = if(n<2, 0, (3*n)!/((2*n+1) * (2*n-1) * (n+1)! * n!^2))}; /* Michael Somos, Jul 04 2020 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Dec 24 1999
a(16)-a(18) corrected and a(19)-a(20) added by Alois P. Heinz, Aug 22 2016
STATUS
approved