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

Author

Marks R. Nester

Keywords

• nonn

Comments

Turning over the necklace is not allowed.

Examples

			For n=5, the 24 necklaces are A followed by the 24 permutations of BCDE.
		

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

Crossrefs

Cf. A000031, A001867, A001868, A001869, A008277.

Column k=5 of A087854.

Programs

Mathematica

k=5; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

PARI

a(n) = my(k=5); k!*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2))/n; \\ Michel Marcus, Sep 27 2018

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)