A215557 Number of standard Young tableaux of shape [n^2,n].

1, 1, 9, 154, 3705, 115101, 4395118, 199448964, 10495906641, 628737007195, 42254306265171, 3148956023335200, 257758558133120135, 22991045919047089170, 2219652431230209792300, 230617851021799852486856, 25657807699789594931790369, 3043509929953923167586547335

Offset: 0

Author

Alois P. Heinz, Aug 16 2012

Keywords

• nonn
• changed

Comments

Also the number of binary words with n^2 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(2) = 9 words are: 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..337
• Wikipedia, Young tableau.

Crossrefs

Main diagonal of A214776.

Programs

Maple

a:= n-> binomial((n+1)*n, n)*((n-1)*n+1)/(n*n+1):
seq(a(n), n=0..20);

Mathematica

Table[Binomial[n(n+1),n] (n(n-1)+1)/(n^2+1),{n,0,20}] (* Harvey P. Dale, Dec 08 2023 *)

Formula

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

a(n) = A214776(n,n).

a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(n^2-n+1). - Ilya Gutkovskiy, Nov 01 2017

a(n) ~ n^(n-1/2) * exp(n+1/2-1/(6*n)) / sqrt(2*Pi). - Amiram Eldar, Aug 29 2025