A056286 Number of n-bead necklaces with exactly six different colored beads.

0, 0, 0, 0, 0, 120, 2160, 23940, 211680, 1643544, 11748240, 79419180, 516257280, 3262443120, 20193277104, 123071707080, 741419995680, 4427490147480, 26264144909520, 155018841055596, 911509010154720, 5344538384445120, 31272099902089200, 182707081122261480

Offset

1,6

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

an) = A054625(n) - 6*A001869(n) + 15*A001868(n) - 20*A001867(n) + 15*A000031(n) - 6.

From Robert A. Russell, Sep 26 2018: (Start)

a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=6 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=6 is the number of colors. (End)

Example

For n=6, the 120 necklaces are A followed by the 120 permutations of BCDEF.

Mathematica

k=6; Table[k!DivisorSum[n, EulerPhi[#]StirlingS2[n/#, k]&]/n, {n, 1, 30}] (* Robert A. Russell, Sep 26 2018 *)

Crossrefs

Cf. A000031, A001867, A001868, A001869, A054625.

Column k=6 of A087854.

Sequence in context: A144858 A084030 A056291 * A166779 A038745 A267839

Adjacent sequences: A056283 A056284 A056285 * A056287 A056288 A056289

Keyword

nonn

Author

Marks R. Nester

Status

approved