A056284 Number of n-bead necklaces with exactly four different colored beads.

0, 0, 0, 6, 48, 260, 1200, 5106, 20720, 81876, 318000, 1223136, 4675440, 17815020, 67769552, 257700906, 980240880, 3731753180, 14222737200, 54278580036, 207438938000, 793940475900, 3043140078000, 11681057249536, 44900438149296, 172824331826580, 666070256489680

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

Turning over the necklace is not allowed.

Examples

			For n=4, the six necklaces are ABCD, ABDC, ACBD, ACDB, ADBC and ADCB.
		

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. A001868.

Column k=4 of A087854.

Programs

Mathematica

k=4; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

PARI

a(n) = my(k=4);(k!/n)*sumdiv(n, d, eulerphi(d)*stirling(n/d,k,2)); \\ Michel Marcus, Sep 27 2018

Formula

a(n) = A001868(n) - 4*A001867(n) + 6*A000031(n) - 4.

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

a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=4 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=4 is the number of colors. (End)