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Even minus odd extensions of truncated 3 X 2n grid diagram.
6

%I #33 Jul 07 2023 13:15:27

%S 2,12,110,1274,17136,255816,4124406,70549050,1264752060,23555382240,

%T 452806924752,8939481277552,180551099694400,3719061442253520,

%U 77933728043586630,1658001861319441050,35749633305661575300,780123576993991461000,17208112644166765652100

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

%C 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

%C From _Joel B. Lewis_, Oct 05 2009: (Start)

%C Also the number of standard shifted Young tableaux of shape (n+1, n, n-1).

%C Also the number of 2143-avoiding up-down permutations of length 2n - 1. (End)

%H J. B. Lewis, <a href="https://dspace.mit.edu/handle/1721.1/73444">Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux</a>, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.

%H F. Ruskey, <a href="http://dx.doi.org/10.1016/0095-8956(92)90067-8">Generating linear extensions of posets by transpositions</a>, J. Combin. Theory, B 54 (1992), 77-101.

%H Dennis White, <a href="http://www.math.umn.edu/~white/papers/index.html">Sign-balanced posets</a>

%F a(n) = multinomial(3n; n-1, n, n+1)/(n(2n-1)(2n+1)).

%F a(n) ~ 3^(3*n+1/2) / (8*Pi*n^4). - _Vaclav Kotesovec_, Nov 13 2014

%F 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

%p A007724 := proc(n)

%p combinat[multinomial](3*n,n-1,n,n+1)/n/(2*n-1)/(2*n+1) ;

%p end proc:

%p seq(A007724(n),n=2..40) ; # _R. J. Mathar_, Jul 07 2023

%t Table[(3*n)!/((n-1)!*n!*(n+1)!)/(n*(2*n-1)*(2*n+1)),{n,2,10}] (* _Vaclav Kotesovec_, Nov 13 2014 *)

%t Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3}, 1], {n, 19}] (* _Michael De Vlieger_, Aug 22 2016 *)

%o (PARI) {a(n) = if(n<2, 0, (3*n)!/((2*n+1) * (2*n-1) * (n+1)! * n!^2))}; /* _Michael Somos_, Jul 04 2020 */

%Y Cf. A003121.

%Y 2143-avoiding up-down permutations of length 2n are given by A005789. - _Joel B. Lewis_, Oct 05 2009

%Y After corrections, is very similar to A217800.

%K nonn,easy

%O 2,1

%A _Frank Ruskey_

%E More terms from _James A. Sellers_, Dec 24 1999

%E a(16)-a(18) corrected and a(19)-a(20) added by _Alois P. Heinz_, Aug 22 2016