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A032194
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Number of necklaces with 9 black beads and n-9 white beads.
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1
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1, 1, 5, 19, 55, 143, 335, 715, 1430, 2704, 4862, 8398, 14000, 22610, 35530, 54484, 81719, 120175, 173593, 246675, 345345, 476913, 650325, 876525, 1168710, 1542684, 2017356, 2615104, 3362260, 4289780, 5433736, 6835972
(list; graph; refs; listen; history; internal format)
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OFFSET
| 9,3
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COMMENTS
| The g.f. is Z(C_9,x)/x^9, the 9-variate cycle index polynomial for the cyclic group C_9, with substitution x[i]->1/(1-x^i), i=1,...,9. Therefore by Polya enumeration a(n+9) is the number of cyclically inequivalent 9-necklaces whose 9 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_9,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[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of 1, 1, 1, 1...
G.f.: (x^9)*(1-5*x+14*x^2-18*x^3+21*x^4-21*x^5+25*x^6-21*x^7+21*x^8-18*x^9+14*x^10-5*x^11+x^12)/((1-x)^6*(1-x^3)^2*(1-x^9)).
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MATHEMATICA
| k = 9; 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, A032193.
Sequence in context: A055365 A060100 A053733 * A024532 A036421 A202658
Adjacent sequences: A032191 A032192 A032193 * A032195 A032196 A032197
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net)
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