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A057599
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a(n) = (n^2)!/(n!)^(n+1); number of ways of dividing n^2 labeled items into n unlabeled boxes of n items each.
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11
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1, 1, 3, 280, 2627625, 5194672859376, 3708580189773818399040, 1461034854396267778567973305958400, 450538787986875167583433232345723106006796340625, 146413934927214422927834111686633731590253260933067148964500000000
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OFFSET
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0,3
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COMMENTS
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Note that if n=p^k for p prime then a(n) is coprime to n (i.e., a(n) is not divisible by p).
a(n) is also the number of labelings for the simple graph K_n X K_n, the graph Cartesian product of the complete graph with itself. - Geoffrey Critzer, Oct 16 2016
a(n) is also the number of knockout tournament seedings with 2 rounds and n participants in each match. - Alexander Karpov, Dec 15 2017
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LINKS
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FORMULA
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a(n) ~ exp(n - 1/12) * n^((n-1)*(2*n-1)/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Nov 23 2018
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EXAMPLE
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a(2)=3 since the possibilities are {{0,1},{2,3}}; {{0,2},{1,3}}; and {{0,3},{1,2}}.
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MAPLE
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a:= n-> (n^2)!/(n!)^(n+1):
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MATHEMATICA
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Table[a[z_] := z^n/n!; (n^2)! Coefficient[Series[a[a[z]], {z, 0, n^2}], z^(n^2)], {n, 1, 10}] (* Geoffrey Critzer, Oct 16 2016 *)
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PROG
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(PARI) a(n) = (n^2)!/(n!)^(n+1); \\ Altug Alkan, Dec 17 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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