A056347 Number of primitive (period n) bracelets using a maximum of six different colored beads.

6, 15, 50, 210, 882, 4220, 20640, 107100, 563730, 3036411, 16514100, 90778485, 502474350, 2799199380, 15673672238, 88162569180, 497847963690, 2821127257950, 16035812864940, 91404065292036

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

Turning over will not create a new bracelet.

References

• M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Column 6 of A276550.

Cf. A032164.

Programs

Mathematica

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}]; CoefficientList[Series[gf[x,6],{x,0,mx}],x] (* Herbert Kociemba, Nov 28 2016 *)

Formula

sum mu(d)*A056341(n/d) where d|n.

From Herbert Kociemba, Nov 28 2016: (Start)

More generally, gf(k) is the g.f. for the number of bracelets with primitive period n and 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)