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A057599 a(n) = (n^2)!/(n!)^(n+1); number of ways of dividing n^2 labeled items into n unlabeled boxes of n items each. 11
1, 1, 3, 280, 2627625, 5194672859376, 3708580189773818399040, 1461034854396267778567973305958400, 450538787986875167583433232345723106006796340625, 146413934927214422927834111686633731590253260933067148964500000000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
LINKS
Alexander Karpov, Generalized knockout tournaments, National Research University Higher School of Economics. WP7/2017/03.
FORMULA
a(n) = A034841(n)/A000142(n).
a(n) ~ exp(n - 1/12) * n^((n-1)*(2*n-1)/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Nov 23 2018
EXAMPLE
a(2)=3 since the possibilities are {{0,1},{2,3}}; {{0,2},{1,3}}; and {{0,3},{1,2}}.
MAPLE
a:= n-> (n^2)!/(n!)^(n+1):
seq(a(n), n=0..10); # Alois P. Heinz, Apr 29 2020
MATHEMATICA
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 *)
PROG
(PARI) a(n) = (n^2)!/(n!)^(n+1); \\ Altug Alkan, Dec 17 2017
CROSSREFS
Main diagonal of A060540.
Sequence in context: A364617 A263884 A096126 * A239273 A054583 A139984
KEYWORD
nonn
AUTHOR
Henry Bottomley, Oct 06 2000
STATUS
approved

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Last modified June 19 15:41 EDT 2024. Contains 373503 sequences. (Running on oeis4.)