A007724 Even minus odd extensions of truncated 3 X 2n grid diagram.

2, 12, 110, 1274, 17136, 255816, 4124406, 70549050, 1264752060, 23555382240, 452806924752, 8939481277552, 180551099694400, 3719061442253520, 77933728043586630, 1658001861319441050, 35749633305661575300, 780123576993991461000, 17208112644166765652100

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

Table of n, a(n) for n=2..20.

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

Cf. A003121.

2143-avoiding up-down permutations of length 2n are given by A005789. - Joel B. Lewis, Oct 05 2009

After corrections, is very similar to A217800.

Sequence in context: A179493 A193268 A235601 * A217800 A241958 A217802

Adjacent sequences: A007721 A007722 A007723 * A007725 A007726 A007727

Keyword

nonn,easy

Author

Frank Ruskey

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