A032197 Number of necklaces with 12 black beads and n-12 white beads.

1, 1, 7, 31, 116, 364, 1038, 2652, 6310, 14000, 29414, 58786, 112720, 208012, 371516, 643856, 1086601, 1789515, 2883289, 4552275, 7056280, 10752060, 16128424, 23841480, 34769374, 50067108, 71250060, 100276894, 139672312

Offset: 12

Author

Christian G. Bower

Keywords

• nonn

Comments

The g.f. is Z(C_12,x)/x^12, the 12-variate cycle index polynomial for the cyclic group C_12, with substitution x[i]->1/(1-x^i), i=1,...,12. Therefore by Polya enumeration a(n+12) is the number of cyclically inequivalent 12-necklaces whose 12 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_12,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

Links

• 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 necklaces
• Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,-4,12,-12,2,2,-12,24,-18,4,4,-6,15,-20,0,10,-4,10,0,-20,15,-6,4,4,-18,24,-12,2,2,-12,12,-4,4,-2,-4,4,-1).

Crossrefs

Column k=12 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192-A032196.

Programs

Mathematica

k = 12; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

Formula

"CIK[ 12 ]" (necklace, indistinct, unlabeled, 12 parts) transform of 1, 1, 1, 1...

G.f.: (x^12)*(1-3*x+7*x^2+9*x^3+18*x^4+38*x^5+72*x^6+92*x^7+168*x^8+160*x^9+238*x^10+230*x^11+296*x^12+234*x^13+330*x^14+248*x^15+284*x^16+238*x^17+230*x^18+166*x^19+172*x^20+78*x^21+80*x^22+38*x^23+21*x^24+7*x^25+3*x^26+x^27) /((1+x)*(1-x)*(1-x^2)*(1-x^3)*(1-x)^5*(1+x+x^2)*(1-x^4)^2*(1-x^6)*(1-x^12)). - Wolfdieter Lang, Feb 15 2005 (see comment)

G.f.: 1/12 x^12 ((1 - x)^-12 + (1 - x^2)^-6 + 2 (1 - x^3)^-4 + 2 (1 - x^4)^-3 + 2 (1 - x^6)^-2 + 4 (1 - x^12)^-1). - Herbert Kociemba, Oct 22 2016