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 A056285 Number of n-bead necklaces with exactly five different colored beads. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Turning over the necklace is not allowed. REFERENCES 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] LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 FORMULA a(n) = A001869(n) - 5*A001868(n) + 10*A001867(n) - 10*A000031(n) + 5. From Robert A. Russell, Sep 26 2018: (Start) 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) EXAMPLE For n=5, the 24 necklaces are A followed by the 24 permutations of BCDE. MATHEMATICA k=5; Table[k!DivisorSum[n, EulerPhi[#]StirlingS2[n/#, k]&]/n, {n, 1, 30}] (* Robert A. Russell, Sep 26 2018 *) PROG (PARI) a(n) = my(k=5); k!*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2))/n; \\ Michel Marcus, Sep 27 2018 CROSSREFS Cf. A000031, A001867, A001868, A001869, A008277. Column k=5 of A087854. Sequence in context: A153782 A073990 A056290 * A162686 A010976 A100130 Adjacent sequences:  A056282 A056283 A056284 * A056286 A056287 A056288 KEYWORD nonn AUTHOR STATUS approved

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Last modified December 13 17:24 EST 2018. Contains 318086 sequences. (Running on oeis4.)