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A032195
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Number of necklaces with 10 black beads and n-10 white beads.
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1
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1, 1, 6, 22, 73, 201, 504, 1144, 2438, 4862, 9252, 16796, 29414, 49742, 81752, 130752, 204347, 312455, 468754, 690690, 1001603, 1430715, 2016144, 2804880, 3856892, 5245128, 7060984, 9414328, 12440668, 16301164
(list; graph; refs; listen; history; internal format)
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OFFSET
| 10,3
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COMMENTS
| The g.f. is Z(C_10,x)/x^10, the 10-variate cycle index polynomial for the cyclic group C_10, with substitution x[i]->1/(1-x^i), i=1,...,10. By Polya enumeration, a(n+10) is the number of cyclically inequivalent 10-necklaces whose 10 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_10,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[ 10 ]" (necklace, indistinct, unlabeled, 10 parts) transform of 1, 1, 1, 1...
G.f.: (x^10)*(1-3*x+4*x^2+12*x^3-8*x^4-x^5+31*x^6-4*x^8+16*x^9+11*x^10+3*x^11+8*x^12+4*x^13+4*x^14+x^15+x^16)/((1-x)^4*(1-x^2)^4*(1-x^5)*(1-x^10)).
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MATHEMATICA
| k = 10; 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, A032194.
Sequence in context: A171495 A178706 A159555 * A111566 A200052 A051945
Adjacent sequences: A032192 A032193 A032194 * A032196 A032197 A032198
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KEYWORD
| nonn
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AUTHOR
| Christian G. Bower (bowerc(AT)usa.net)
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