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A032196
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Number of necklaces with 11 black beads and n-11 white beads.
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1
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1, 1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32066, 58786, 104006, 178296, 297160, 482885, 766935, 1193010, 1820910, 2731365, 4032015, 5864750, 8414640, 11920740, 16689036, 23107896, 31666376, 42975796
(list; graph; refs; listen; history; internal format)
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OFFSET
| 11,3
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COMMENTS
| The g.f. is Z(C_11,x)/x^11, the 11-variate cycle index polynomial for the cyclic group C_11, with substitution x[i]->1/(1-x^i), i=1,...,11. By Polya enumeration, a(n+11) is the number of cyclically inequivalent 11-necklaces whose 11 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_11,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 15 2005.
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LINKS
| C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Index entries for sequences related to necklaces
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FORMULA
| "CIK[ 11 ]" (necklace, indistinct, unlabeled, 11 parts) transform of 1, 1, 1, 1...
G.f.:(x^11)*(1-9*x+41*x^2-109*x^3+191*x^4-229*x^5+191*x^6-109*x^7+41*x^8-9*x^9+x^10 )/((1-x)^10*(1-x^11)).
a(n)=ceiling(binomial(n+11, 11)/(n+11)) (conjecture W. Lang).
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MATHEMATICA
| k = 11; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] - Robert A. Russell (russell(AT)post.harvard.edu), Sep 27 2004
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CROSSREFS
| Cf. A004526, A007997, A008610, A008646, A032191, A032192-A032195.
Sequence in context: A036422 A166214 A032169 * A011780 A036631 A036638
Adjacent sequences: A032193 A032194 A032195 * A032197 A032198 A032199
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net)
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