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%I #24 Sep 28 2018 10:06:06
%S 0,0,0,0,24,300,2400,15750,92680,510312,2691600,13794150,69309240,
%T 343501500,1686135376,8221437000,39901776360,193054016840,
%U 932142850800,4495236798162,21664357535320,104388120866100,503044634004000,2425003924383900,11696087875731624
%N Number of n-bead necklaces with exactly five 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="/A056285/b056285.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A001869(n) - 5*A001868(n) + 10*A001867(n) - 10*A000031(n) + 5.
%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=5 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=5 is the number of colors. (End)
%e For n=5, the 24 necklaces are A followed by the 24 permutations of BCDE.
%t k=5; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* _Robert A. Russell_, Sep 26 2018 *)
%o (PARI) a(n) = my(k=5); k!*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2))/n; \\ _Michel Marcus_, Sep 27 2018
%Y Cf. A000031, A001867, A001868, A001869, A008277.
%Y Column k=5 of A087854.
%K nonn
%O 1,5
%A _Marks R. Nester_
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