cp's OEIS Frontend

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

%I A215557 #27 Aug 29 2025 04:15:49
%S A215557 1,1,9,154,3705,115101,4395118,199448964,10495906641,628737007195,
%T A215557 42254306265171,3148956023335200,257758558133120135,
%U A215557 22991045919047089170,2219652431230209792300,230617851021799852486856,25657807699789594931790369,3043509929953923167586547335
%N A215557 Number of standard Young tableaux of shape [n^2,n].
%C A215557 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.
%H A215557 Alois P. Heinz, <a href="/A215557/b215557.txt">Table of n, a(n) for n = 0..337</a>
%H A215557 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>.
%F A215557 a(n) = C((n+1)*n, n)*((n-1)*n+1)/(n*n+1).
%F A215557 a(n) = A214776(n,n).
%F A215557 a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(n^2-n+1). - _Ilya Gutkovskiy_, Nov 01 2017
%F A215557 a(n) ~ n^(n-1/2) * exp(n+1/2-1/(6*n)) / sqrt(2*Pi). - _Amiram Eldar_, Aug 29 2025
%p A215557 a:= n-> binomial((n+1)*n, n)*((n-1)*n+1)/(n*n+1):
%p A215557 seq(a(n), n=0..20);
%t A215557 Table[Binomial[n(n+1),n] (n(n-1)+1)/(n^2+1),{n,0,20}] (* _Harvey P. Dale_, Dec 08 2023 *)
%Y A215557 Main diagonal of A214776.
%K A215557 nonn,changed
%O A215557 0,3
%A A215557 _Alois P. Heinz_, Aug 16 2012