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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
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
Column k=5 of A087854.
Sequence in context: A153782 A073990 A056290 * A162686 A010976 A100130
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)