A252765 - OEIS

%I #14 Mar 25 2017 06:39:35

%S 1,1,2,188,3941598,24934429725024,74171603795480180204640,

%T 150277870737901828652705825755721760,

%U 283839436431731355577562936415156522873876247241520,655934428473920614716696820356119117524334608980167506174657536026880

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

%H Alois P. Heinz, <a href="/A252765/b252765.txt">Table of n, a(n) for n = 0..25</a>

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

%F From _Vaclav Kotesovec_, Aug 23 2015: (Start)

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

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

%F (End)

%p with(numtheory):

%p a:= n-> `if`(n=0, 1, add(phi(n/d)*(n*d)!/(d!^n*n^2), d=divisors(n))):

%p seq(a(n), n=0..10);

%t a[n_] := If[n == 0, 1, DivisorSum[n, EulerPhi[n/#]*(n*#)!/(#!^n*n^2)&]];

%t Table[a[n], {n, 0, 10}] (* _Jean-François Alcover_, Mar 25 2017, translated from Maple *)

%Y Main diagonal of A208183.

%Y Cf. A000010, A000142.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Dec 21 2014