gefundenes Fressen 3, and cured of 'm.

[wouter meeussen]

neclaces are elegant, but bracelets look gross:

consider the table of bracelet polynomials, n beads, c<=n colours,
substitute colour k by Exp[2 Pi I k/c],
Table[NecklacePolynomial[n,Exp[2 I Pi
Range[c]/c],Dihedral]//FullSimplify,{n,13},{c,n}]
and find:
{1},
{1,1},
{1,0,1},
{1,2,0,1},
{1,0,0,0,2},
{1,3,2,0,0,1},
{1,0,0,0,0,0,3},
{1,6,0,3,0,0,0,2},
{1,0,4,0,0,0,0,0,3},
{1,10,0,0,6,0,0,0,0,2},
{1,0,0,0,0,0,0,0,0,0,5},
{1,20,8,6,0,4,0,0,0,0,0,2},
{1,0,0,0,0,0,0,0,0,0,0,0,6},

and that might very well be:
if c|n then if(c=1,2,1)*(1/2/c)*
          (if(c=2,2^(n/c),0)+ Sum(d|(n/c), phi(c*d)*c^(n/c/d)/(n/c))
else 0

or,

Table[If[MemberQ[Divisors[n],c],If[c===1,2,1]*(1/2/c)
*(If[c===2,2^(n/c),0]+Fold[#1+ EulerPhi[c #2]*c^(n/#2/c)
/(n/c)&,0,Divisors[n/c]]),0],{n,13},{c,n}]

which is not nice at all, with ugly If's showing definite lack of elegance
(?!)
Anyone to de-grossify this?
But I honestly admit still not to grasp (=grok) the 'why' of these triviae.
So, I give up.

Wouter.