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A019536
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a(n) = number of necklaces of n beads with up to n unlabeled colors.
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2
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1, 2, 5, 20, 109, 784, 6757, 68240, 787477, 10224812, 147512053, 2340964372, 40527565261, 760095929840, 15352212731933, 332228417657960, 7668868648772701, 188085259070219000, 4884294069438337429
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Moebius transform of a(n) equals A060223. Possibly identical to A019536.
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LINKS
| M. Goebel, On the number of special permutation-invariant orbits and terms, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.)
F. Ruskey, Necklaces with Fixed Density
Eric Weisstein's world of Mathematics, Necklaces
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FORMULA
| See Mathematica code
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EXAMPLE
| a(3)=5 since the partitions of the 3 (unlabeled) colors are {3}:RRR, {2,1}:RRG and {1,1,1}:RGB, with multiplicities resp. *1, *2, *1, ( *2 since partitioning 3 beads over a bin of 2 and a bin of 1), so the necklaces are {R,R,R},{R,R,G},{R,G,G},{R,G,B},{R,B,G}.
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MATHEMATICA
| Needs["DiscreteMath`Combinatorica`"]; mult[li:{__Integer}] := Multinomial @@ Length /@ Split[Sort[li]]; neck[li:{__Integer}] := Module[{n, d}, n=Plus @@ li; d=n-First[li]; Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times @@ ((li/#2)!)&, 0, Divisors[GCD @@ li]]/n]; Table[(mult /@ Partitions[n]).(neck /@ Partitions[n]), {n, 24}]
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CROSSREFS
| Cf. A000670.
Row sums of A087854. - DELEHAM Philippe.
Sequence in context: A152562 A006867 A170946 * A129949 A127065 A168357
Adjacent sequences: A019533 A019534 A019535 * A019537 A019538 A019539
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KEYWORD
| easy,nonn
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AUTHOR
| Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)
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EXTENSIONS
| Edited by Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 06 2002
Corrected by T. D. Noe (noe(AT)sspectra.com), Oct 31 2006
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