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A207816 Number of distinct necklaces with n red, n green, n blue and n white beads.

%I A207816 #31 Mar 24 2017 03:48:29
%S A207816 1,6,318,30804,3941598,586637256,96197661156,16875655269948,
%T A207816 3111284141045598,595909785174057204,117634021777132574568,
%U A207816 23797087019979071174580,4912693780461352534397604,1031629572413246016139181544,219809927417367534490107035244,47426945434432859336092700072304
%N A207816 Number of distinct necklaces with n red, n green, n blue and n white beads.
%H A207816 Alois P. Heinz, <a href="/A207816/b207816.txt">Table of n, a(n) for n = 0..400</a>
%F A207816 a(n) = Sum_{d|n} phi(n/d)*(4*d)!/(d!^4*4*n) if n>0 and a(0) = 1. - _Alois P. Heinz_, Feb 24 2012
%F A207816 a(n) ~ 2^(8*n-5/2) / (Pi^(3/2) * n^(5/2)). - _Vaclav Kotesovec_, Aug 23 2015
%e A207816 For n=1, a(1)=6 since for four beads necklaces with each bead from each of the four colors say (R,G,B,W), we can arrange as following, [R,G,B,W], [R,G,W,B], [R,B,G,W], [R,B,W,G], [R,W,G,B] and [R,W,B,G].
%p A207816 with(combinat): with(numtheory):
%p A207816 # This formula comes from Polya Counting Theorem:
%p A207816 # Z(C_n) = add(phi(d)*(a_d)^(n/d), d in divisors(n))/n;
%p A207816 PolyaBrace:= proc(S) option remember; local n, s, d;
%p A207816                n:= add(s, s=S);
%p A207816                add(phi(d) *PolyaCoeff(d, S), d=divisors(n))/n
%p A207816              end:
%p A207816 # Find coeff of prod(a[i]^s[i], i=1..n) of a_d^(n/d) (symmetric function)
%p A207816 PolyaCoeff:= proc(d, S) option remember; local n, pow, s;
%p A207816                n:= add(s, s=S);
%p A207816                pow:= n/d;
%p A207816                if {seq(s mod d, s = S)} = {0}
%p A207816                   then multinomial(pow, seq(s/d, s = S))
%p A207816                   else 0
%p A207816                fi:
%p A207816              end:
%p A207816 a:= n-> `if`(n=0, 1, PolyaBrace([n$4])):
%p A207816 seq(a(n), n=0..20);
%t A207816 a[n_] := DivisorSum[n, EulerPhi[n/#] (4#)!/(#!^4 * 4n)&]; a[0]=1;
%t A207816 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 24 2017, after _Alois P. Heinz_ *)
%Y A207816 Cf. A003239, A118644, A137729.
%Y A207816 Column k=4 of A208183. - _Alois P. Heinz_, Feb 24 2012
%K A207816 nonn
%O A207816 0,2
%A A207816 _Thotsaporn Thanatipanonda_, Feb 20 2012