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%I #20 Sep 28 2018 09:59:34
%S 0,0,0,0,0,120,2160,23940,211680,1643544,11748240,79419180,516257280,
%T 3262443120,20193277104,123071707080,741419995680,4427490147480,
%U 26264144909520,155018841055596,911509010154720,5344538384445120,31272099902089200,182707081122261480
%N Number of n-bead necklaces with exactly six different colored beads.
%C Turning over the necklace is not allowed.
%D 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]
%H Alois P. Heinz, <a href="/A056286/b056286.txt">Table of n, a(n) for n = 1..1000</a>
%F an) = A054625(n) - 6*A001869(n) + 15*A001868(n) - 20*A001867(n) + 15*A000031(n) - 6.
%F From _Robert A. Russell_, Sep 26 2018: (Start)
%F 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.
%F 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)
%e For n=6, the 120 necklaces are A followed by the 120 permutations of BCDEF.
%t k=6; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* _Robert A. Russell_, Sep 26 2018 *)
%Y Cf. A000031, A001867, A001868, A001869, A054625.
%Y Column k=6 of A087854.
%K nonn
%O 1,6
%A _Marks R. Nester_
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