A214955 Number of solid standard Young tableaux of shape [[n,n-1],[1]].

1, 6, 25, 98, 378, 1452, 5577, 21450, 82654, 319124, 1234506, 4784276, 18572500, 72209880, 281150505, 1096087770, 4278278070, 16717354500, 65388738030, 256000696380, 1003116947820, 3933750236520, 15437682614250, 60625494924228, 238235373671148, 936735006679752

Offset: 1

Author

Alois P. Heinz, Jul 30 2012

Keywords

• nonn

Comments

a(n) is odd if and only if n = 2^i-1 for i in {1, 2, 3, ...} = A000027.

Form an array with m(1,n) = n*(n+1)/2, m(n,1) = n*(n-1)+1, and m(i,j) = m(i,j-1) + m(i-1,j); A000217 in the top row, A002061 in the first column, A086514 in the second column. Then on the diagonal m(n,n) = a(n). - J. M. Bergot, May 02 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012.
• Wikipedia, Young tableau.

Crossrefs

Column k=1 of A214775.

Cf. A000027, A000108, A006752, A060747, A215002.

Programs

Maple

a:= proc(n) option remember;
      `if`(n<2, n, 2*(2*n-1)^2*a(n-1)/((n+1)*(2*n-3)))
    end:
seq(a(n), n=1..30);

Mathematica

a[n_]:= a[n] = If[n<2, n, 2*(2*n-1)^2*a[n-1]/((n+1)*(2*n-3))]; Array[a, 30] (* Jean-François Alcover, Aug 14 2017, translated from Maple *)

PARI

a(n) = (2*n-1) * binomial(2*n,n)/(n+1); \\ Michel Marcus, Mar 06 2022

Formula

a(n) = 2*(2*n-1)^2/((n+1)*(2*n-3)) * a(n-1) for n>1; a(1) = 1.

a(n) = (2*n-1) * C(2*n,n)/(n+1) = A060747(n) * A000108(n).

a(n) = [x^n] x*(1 + 2*x)/(1 - x)^(n+2). - Ilya Gutkovskiy, Oct 12 2017

Sum_{n>=1} 1/a(n) = 1/6 + G + 13*Pi/(36*sqrt(3)) - Pi*log(2+sqrt(3))/8, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 06 2022

From Stefano Spezia, Mar 29 2023: (Start)

O.g.f.: 1 + (3 - 3*sqrt(1 - 4*x) - 8*x)/(2*x*sqrt(1 - 4*x)).

E.g.f.: 1 + exp(2*x)*(3*I_1(2*x) - I_0(2*x)), where I_n(x) is the modified Bessel function of the first kind.

a(n) ~ 2^(1+2*n)/sqrt(n*Pi). (End)