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A056285
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Number of n-bead necklaces with exactly five different colored beads.
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7
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0, 0, 0, 0, 24, 300, 2400, 15750, 92680, 510312, 2691600, 13794150, 69309240, 343501500, 1686135376, 8221437000, 39901776360, 193054016840, 932142850800, 4495236798162, 21664357535320, 104388120866100, 503044634004000, 2425003924383900, 11696087875731624
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OFFSET
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1,5
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COMMENTS
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Turning over the necklace is not allowed.
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REFERENCES
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M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
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LINKS
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FORMULA
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a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=5 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=5 is the number of colors. (End)
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EXAMPLE
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For n=5, the 24 necklaces are A followed by the 24 permutations of BCDE.
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MATHEMATICA
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k=5; Table[k!DivisorSum[n, EulerPhi[#]StirlingS2[n/#, k]&]/n, {n, 1, 30}] (* Robert A. Russell, Sep 26 2018 *)
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PROG
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(PARI) a(n) = my(k=5); k!*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2))/n; \\ Michel Marcus, Sep 27 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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