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A032242 Number of identity bracelets of n beads of 5 colors. 1
5, 10, 10, 45, 252, 1120, 5270, 23475, 106950, 483504, 2211650, 10148630, 46911060, 217863040, 1017057256, 4767774375, 22438419120, 105960830300, 501928967930, 2384170903140, 11353241255900 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..21.

C. G. Bower, Transforms (2)

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

Index entries for sequences related to bracelets

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)

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 *)

CROSSREFS

Sequence in context: A201033 A242894 A256641 * A208541 A107975 A262665

Adjacent sequences:  A032239 A032240 A032241 * A032243 A032244 A032245

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified October 19 00:36 EDT 2018. Contains 316327 sequences. (Running on oeis4.)