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A261599 Number of primitive (aperiodic, or Lyndon) necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct. 4
1, 1, 0, 1, 1, 3, 13, 24, 67, 252, 1795, 4038, 16812, 61750, 349806, 3485026, 10391070, 49433135, 240064988, 1282012986, 9167581934, 131550811985, 459677212302, 2707382738558, 14318807586215, 94084166753923, 601900541189696, 5894253303715121 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..300

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]

Eric Weisstein's World of Mathematics, Necklace

Wikipedia, Lyndon word

Wikipedia, Necklace (combinatorics)

Index entries for sequences related to necklaces

FORMULA

a(n) = (1/n) * Sum_{d | n} moebius(n/d) * A007837(d) for n>0. - Andrew Howroyd, Dec 21 2017

EXAMPLE

a(4) = 1: 0001.

a(5) = 3: 00001, 00011, 00101.

a(6) = 13: 000001, 000011, 000101, 000112, 000121, 000122, 001012, 001021, 001022, 001102, 001201, 001202, 010102.

a(7) = 24: 0000001, 0000011, 0000101, 0000111, 0000112, 0000121, 0000122, 0001001, 0001011, 0001012, 0001021, 0001022, 0001101, 0001102, 0001201, 0001202, 0010011, 0010012, 0010021, 0010022, 0010101, 0010102, 0010201, 0010202.

MAPLE

with(numtheory):

b:= proc(n, i, g, d, j) option remember; `if`(i*(i+1)/2<n or g>0

       and g<d, 0, `if`(n=0, `if`(d=g, 1, 0), b(n, i-1, g, d, j)+

      `if`(i>n, 0, binomial(n/j, i/j)*b(n-i, i-1, igcd(i, g), d, j))))

    end:

a:= n-> `if`(n=0, 1, add(add((f-> `if`(f=0, 0, f*b(n$2, 0, d, j)))(

                     mobius(j)), j=divisors(d)), d=divisors(n))/n):

seq(a(n), n=0..30);

MATHEMATICA

a[0] = 1; a[n_] := With[{P = Product[1 + x^k/k!, {k, 1, n}] + O[x]^(n+1) // Normal}, DivisorSum[n, MoebiusMu[n/#]*#!*Coefficient[P, x, #]&]/n];

Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, May 28 2018, after Andrew Howroyd *)

PROG

(PARI) a(n)={if(n==0, 1, my(p=prod(k=1, n, (1+x^k/k!) + O(x*x^n))); sumdiv(n, d, moebius(n/d)*d!*polcoeff(p, d))/n)} \\ Andrew Howroyd, Dec 21 2017

CROSSREFS

Cf. A007837, A261531, A261600.

Sequence in context: A260798 A102010 A121718 * A144826 A030552 A146371

Adjacent sequences:  A261596 A261597 A261598 * A261600 A261601 A261602

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 25 2015

STATUS

approved

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Last modified October 28 13:26 EDT 2020. Contains 338055 sequences. (Running on oeis4.)