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%I #20 Sep 28 2018 10:00:41
%S 0,0,0,6,48,260,1200,5106,20720,81876,318000,1223136,4675440,17815020,
%T 67769552,257700906,980240880,3731753180,14222737200,54278580036,
%U 207438938000,793940475900,3043140078000,11681057249536,44900438149296,172824331826580,666070256489680
%N Number of n-bead necklaces with exactly four 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="/A056284/b056284.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A001868(n) - 4*A001867(n) + 6*A000031(n) - 4.
%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=4 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=4 is the number of colors. (End)
%e For n=4, the six necklaces are ABCD, ABDC, ACBD, ACDB, ADBC and ADCB.
%t k=4; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* _Robert A. Russell_, Sep 26 2018 *)
%o (PARI) a(n) = my(k=4);(k!/n)*sumdiv(n, d, eulerphi(d)*stirling(n/d,k,2)); \\ _Michel Marcus_, Sep 27 2018
%Y Cf. A001868.
%Y Column k=4 of A087854.
%K nonn
%O 1,4
%A _Marks R. Nester_
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