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

Author

Frank Ruskey

Keywords

• nonn
• easy

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

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.

Programs

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 *)

PARI

{a(n) = if(n<2, 0, (3*n)!/((2*n+1) * (2*n-1) * (n+1)! * n!^2))}; /* Michael Somos, Jul 04 2020 */

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

Extensions

More terms from James Sellers, Dec 24 1999

a(16)-a(18) corrected and a(19)-a(20) added by Alois P. Heinz, Aug 22 2016