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A215557
Number of standard Young tableaux of shape [n^2,n].
2
1, 1, 9, 154, 3705, 115101, 4395118, 199448964, 10495906641, 628737007195, 42254306265171, 3148956023335200, 257758558133120135, 22991045919047089170, 2219652431230209792300, 230617851021799852486856, 25657807699789594931790369, 3043509929953923167586547335
OFFSET
0,3
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
Wikipedia, Young tableau
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
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 *)
CROSSREFS
Main diagonal of A214776.
Sequence in context: A374854 A173982 A185759 * A365015 A208998 A236578
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 16 2012
STATUS
approved