A214955 - OEIS

%I #68 Jun 05 2023 20:54:57

%S 1,6,25,98,378,1452,5577,21450,82654,319124,1234506,4784276,18572500,

%T 72209880,281150505,1096087770,4278278070,16717354500,65388738030,

%U 256000696380,1003116947820,3933750236520,15437682614250,60625494924228,238235373671148,936735006679752

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

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

%C 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

%H Alois P. Heinz, <a href="/A214955/b214955.txt">Table of n, a(n) for n = 1..1000</a>

%H S. B. Ekhad and D. Zeilberger, <a href="https://arxiv.org/abs/1202.6229">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>, arXiv:1202.6229 [math.CO], 2012.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>.

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

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

%F a(n) = [x^n] x*(1 + 2*x)/(1 - x)^(n+2). - _Ilya Gutkovskiy_, Oct 12 2017

%F 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

%F From _Stefano Spezia_, Mar 29 2023: (Start)

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

%F 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.

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

%p a:= proc(n) option remember;

%p       `if`(n<2, n, 2*(2*n-1)^2*a(n-1)/((n+1)*(2*n-3)))

%p     end:

%p seq(a(n), n=1..30);

%t 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 *)

%o (PARI) a(n) = (2*n-1) * binomial(2*n,n)/(n+1); \\ _Michel Marcus_, Mar 06 2022

%Y Column k=1 of A214775.

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

%K nonn

%O 1,2

%A _Alois P. Heinz_, Jul 30 2012