A208193 Number of distinct 8-colored necklaces with n beads per color.

1, 5040, 5108105520, 15391623287043360, 74701932179186551241520, 474389544274867071519255599040, 3581026866351385580856518554063502880, 30495546426686489361833408314854897254404320, 283839436431731355577562936415156522873876247241520

Offset: 0

Author

Alois P. Heinz, Feb 24 2012

Keywords

• nonn

Comments

In general, column k > 1 of A208183 is asymptotic to k^(k*n-1/2) / ((2*Pi)^((k-1)/2) * n^((k+1)/2)). - Vaclav Kotesovec, Aug 23 2015

Examples

			a(0) =    1: the empty necklace.
a(1) = 5040: {01234567, 01234576, ..., 07654321}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..50

Crossrefs

Column k=8 of A208183.

Cf. A000010, A000142.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(phi(n/d) *(8*d)!/(d!^8 *8*n), d=divisors(n))):
seq(a(n), n=0..10);

Formula

a(n) = Sum_{d|n} phi(n/d)*(8*d)!/(d!^8*8*n) if n>0 and a(0) = 1.

a(n) ~ 8^(8*n-1/2) / ((2*Pi)^(7/2) * n^(9/2)). - Vaclav Kotesovec, Aug 23 2015