A056285 Number of n-bead necklaces with exactly five different colored beads.

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

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

Marks R. Nester

Status

approved