A261531 Number of necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.

1, 1, 1, 2, 2, 4, 15, 25, 69, 254, 1799, 4039, 16828, 61751, 349831, 3485031, 10391139, 49433136, 240065255, 1282012987, 9167583734, 131550812011, 459677216341, 2707382738559, 14318807603110, 94084166753927, 601900541251447, 5894253303715375

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..260

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, Necklace (combinatorics)

Index entries for sequences related to necklaces

Formula

a(n) = (1/n) * Sum_{d | n} phi(n/d) * A007837(d) for n>0. - Andrew Howroyd, Apr 02 2017

Example

a(4) = 2: 0000, 0001.
a(5) = 4: 00000, 00001, 00011, 00101.
a(6) = 15: 000000, 000001, 000011, 000101, 000112, 000121, 000122, 001001, 001012, 001021, 001022, 001102, 001201, 001202, 010102.

Maple

with(numtheory): with(combinat):
g:= l-> (n-> `if`(n=0, 1, add(phi(j)*multinomial(n/j,
        (l/j)[]), j=divisors(igcd(l[])))/n))(add(i, i=l)):
b:= proc(n, i, l) `if`(i*(i+1)/2<n, 0, `if`(n=0, g(l),
      b(n, i-1, l)+`if`(i>n, 0, b(n-i, i-1, [l[], i]))))
    end:
a:= n-> b(n$2, []):
seq(a(n), n=0..35);

Mathematica

multinomial[n_, k_] := n!/Times @@ (k!);
g[l_] := Function[n, If[n==0, 1, Sum[EulerPhi[j]*multinomial[n/j, l/j], {j, Divisors[GCD @@ l]}]/n]][Total[l]];
b[n_, i_, l_] := If[i*(i+1)/2<n, 0, If[n==0, g[l], b[n, i-1, l] + If[i>n, 0, b[n-i, i-1, Append[l, i]]]]];
a[n_] := b[n, n, {}];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

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, eulerphi(n/d)*d!*polcoeff(p, d))/n)} \\ Andrew Howroyd, Dec 21 2017

Crossrefs

Cf. A072605, A261599, A261600.

Sequence in context: A205310 A222924 A268423 * A153968 A153965 A121221

Adjacent sequences: A261528 A261529 A261530 * A261532 A261533 A261534

Keyword

nonn

Author

Alois P. Heinz, Aug 23 2015

Status

approved