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.

1, 1, 3, 280, 2627625, 5194672859376, 3708580189773818399040, 1461034854396267778567973305958400, 450538787986875167583433232345723106006796340625, 146413934927214422927834111686633731590253260933067148964500000000

Offset: 0

Author

Henry Bottomley, Oct 06 2000

Keywords

• nonn

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

Examples

			a(2)=3 since the possibilities are {{0,1},{2,3}}; {{0,2},{1,3}}; and {{0,3},{1,2}}.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..27
• Alexander Karpov, Generalized knockout tournaments, National Research University Higher School of Economics. WP7/2017/03.

Crossrefs

Main diagonal of A060540.

Cf. A000142, A034841.

Programs

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 *)

PARI

a(n) = (n^2)!/(n!)^(n+1); \\ Altug Alkan, Dec 17 2017

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