A032242 Number of identity bracelets of n beads of 5 colors.

5, 10, 10, 45, 252, 1120, 5270, 23475, 106950, 483504, 2211650, 10148630, 46911060, 217863040, 1017057256, 4767774375, 22438419120, 105960830300, 501928967930, 2384170903140, 11353241255900

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Comments

For n>2 also number of asymmetric bracelets with n beads of five colors. - Herbert Kociemba, Nov 29 2016

Links

• Robert Israel, Table of n, a(n) for n = 1..1434
• C. G. Bower, Transforms (2)
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• Index entries for sequences related to bracelets

Crossrefs

Column k=5 of A309528 for n >= 3.

Programs

Maple

N:= 50: # for a(1)..a(N)
G:= add(1/2*numtheory:-mobius(n)*(-log(1-5*x^n)/n - add(binomial(5,i)*x^(n*i)/(1-5*x^(2*n)),i=0..2)), n=1..N):
S:= series(G,x,N+1):
5,10,seq(coeff(S,x,j),j=3..N); # Robert Israel, Jun 24 2019

Mathematica

m=5; (* asymmetric bracelets of n beads of m colors *) Table[Sum[MoebiusMu[d](m^(n/d)/n - If[OddQ[n/d], m^((n/d+1)/2), ((m+1)m^(n/(2d))/2)]), {d,Divisors[n]}]/2, {n,3,20}] (* Robert A. Russell, Mar 18 2013 *)
mx=40;gf[x_,k_]:=Sum[MoebiusMu[n]*(-Log[1-k*x^n]/n-Sum[Binomial[k,i]x^(n i),{i,0,2}]/(1-k x^(2n)))/2,{n,mx}];ReplacePart[Rest[CoefficientList[Series[gf[x,5],{x,0,mx}],x]],{1->5,2->10}] (* Herbert Kociemba, Nov 29 2016 *)

PARI

a(n)={if(n<3, binomial(5,n), sumdiv(n, d, moebius(n/d)*(5^d/n - if(d%2, 5^((d+1)/2), 3*5^(d/2))))/2)} \\ Andrew Howroyd, Sep 12 2019

Formula

"DHK" (bracelet, identity, unlabeled) transform of 5, 0, 0, 0...

From Herbert Kociemba, Nov 29 2016: (Start)

More generally, gf(k) is the g.f. for the number of asymmetric bracelets with n beads of k colors.

gf(k): Sum_{n>=1} mu(n)*( -log(1-k*x^n)/n - Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)) )/2. (End)