OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
FORMULA
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
a(n) ~ 2^(8*n-5/2) / (Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Aug 23 2015
EXAMPLE
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].
MAPLE
with(combinat): with(numtheory):
# This formula comes from Polya Counting Theorem:
# Z(C_n) = add(phi(d)*(a_d)^(n/d), d in divisors(n))/n;
PolyaBrace:= proc(S) option remember; local n, s, d;
n:= add(s, s=S);
add(phi(d) *PolyaCoeff(d, S), d=divisors(n))/n
end:
# Find coeff of prod(a[i]^s[i], i=1..n) of a_d^(n/d) (symmetric function)
PolyaCoeff:= proc(d, S) option remember; local n, pow, s;
n:= add(s, s=S);
pow:= n/d;
if {seq(s mod d, s = S)} = {0}
then multinomial(pow, seq(s/d, s = S))
else 0
fi:
end:
a:= n-> `if`(n=0, 1, PolyaBrace([n$4])):
seq(a(n), n=0..20);
MATHEMATICA
a[n_] := DivisorSum[n, EulerPhi[n/#] (4#)!/(#!^4 * 4n)&]; a[0]=1;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Thotsaporn Thanatipanonda, Feb 20 2012
STATUS
approved