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A032193
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Number of necklaces with 8 black beads and n-8 white beads.
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2
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1, 1, 5, 15, 43, 99, 217, 429, 810, 1430, 2438, 3978, 6310, 9690, 14550, 21318, 30667, 43263, 60115, 82225, 111041, 148005, 195143, 254475, 328756, 420732, 534076, 672452, 840652, 1043460, 1287036, 1577532, 1922741
(list; graph; refs; listen; history; internal format)
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OFFSET
| 8,3
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COMMENTS
| The g.f. is Z(C_8,x)/x^8, the 8-variate cycle index polynomial for the cyclic group C_8, with substitution x[i]->1/(1-x^i), i=1,...,8. Therefore by Polya enumeration a(n+8) is the number of cyclically inequivalent 8-necklaces whose 8 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_8,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[ 8 ]" (necklace, indistinct, unlabeled, 8 parts) transform of 1, 1, 1, 1...
G.f.: (x^8)*(1-3*x+5*x^2+3*x^3-4*x^4+4*x^5+6*x^6-4*x^7+7*x^8-x^9+x^10+x^11)/((1-x)^4*(1-x^2)^2*(1-x^4)*(1-x^8))
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MATHEMATICA
| k = 8; 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.
Sequence in context: A053731 A111295 A200760 * A178965 A005665 A025471
Adjacent sequences: A032190 A032191 A032192 * A032194 A032195 A032196
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net)
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