cp's OEIS Frontend

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

%I A007724 #34 Jul 02 2025 16:01:55
%S A007724 2,12,110,1274,17136,255816,4124406,70549050,1264752060,23555382240,
%T A007724 452806924752,8939481277552,180551099694400,3719061442253520,
%U A007724 77933728043586630,1658001861319441050,35749633305661575300,780123576993991461000,17208112644166765652100
%N A007724 Even minus odd extensions of truncated 3 X 2n grid diagram.
%C A007724 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 A007724 From _Joel B. Lewis_, Oct 05 2009: (Start)
%C A007724 Also the number of standard shifted Young tableaux of shape (n+1, n, n-1).
%C A007724 Also the number of 2143-avoiding up-down permutations of length 2n - 1. (End)
%H A007724 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 A007724 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 A007724 Dennis White, <a href="http://www.math.umn.edu/~white/papers/index.html">Sign-balanced posets</a>
%F A007724 a(n) = multinomial(3n; n-1, n, n+1)/(n(2n-1)(2n+1)).
%F A007724 a(n) ~ 3^(3*n+1/2) / (8*Pi*n^4). - _Vaclav Kotesovec_, Nov 13 2014
%F A007724 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 A007724 := proc(n)
%p A007724     combinat[multinomial](3*n,n-1,n,n+1)/n/(2*n-1)/(2*n+1) ;
%p A007724 end proc:
%p A007724 seq(A007724(n),n=2..40) ; # _R. J. Mathar_, Jul 07 2023
%t A007724 Table[(3*n)!/((n-1)!*n!*(n+1)!)/(n*(2*n-1)*(2*n+1)),{n,2,10}] (* _Vaclav Kotesovec_, Nov 13 2014 *)
%t A007724 Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3}, 1], {n, 19}] (* _Michael De Vlieger_, Aug 22 2016 *)
%o A007724 (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 A007724 Cf. A003121.
%Y A007724 2143-avoiding up-down permutations of length 2n are given by A005789. - _Joel B. Lewis_, Oct 05 2009
%Y A007724 After corrections, is very similar to A217800.
%K A007724 nonn,easy
%O A007724 2,1
%A A007724 _Frank Ruskey_
%E A007724 More terms from _James Sellers_, Dec 24 1999
%E A007724 a(16)-a(18) corrected and a(19)-a(20) added by _Alois P. Heinz_, Aug 22 2016