|
| |
|
|
A032197
|
|
Number of necklaces with 12 black beads and n-12 white beads.
|
|
1
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
12,3
|
|
|
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
|
|
|
|
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
|
|
|
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 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
