A252765 Number of distinct n-colored necklaces with n beads per color.

1, 1, 2, 188, 3941598, 24934429725024, 74171603795480180204640, 150277870737901828652705825755721760, 283839436431731355577562936415156522873876247241520, 655934428473920614716696820356119117524334608980167506174657536026880

Offset: 0

Author

Alois P. Heinz, Dec 21 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..25

Crossrefs

Main diagonal of A208183.

Cf. A000010, A000142.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(phi(n/d)*(n*d)!/(d!^n*n^2), d=divisors(n))):
seq(a(n), n=0..10);

Mathematica

a[n_] := If[n == 0, 1, DivisorSum[n, EulerPhi[n/#]*(n*#)!/(#!^n*n^2)&]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Mar 25 2017, translated from Maple *)

Formula

a(n) = Sum_{d|n} phi(n/d)*(n*d)!/(d!^k*n^2) for n>0, a(0) = 1.

From Vaclav Kotesovec, Aug 23 2015: (Start)

a(n) ~ (n^2)! / (n^2 * (n!)^n).

a(n) ~ n^(n^2 - n/2 - 1) / (exp(1/12) * (2*Pi)^((n-1)/2)).

(End)