%I #37 Sep 04 2018 03:34:29
%S 0,0,1,2,5,18,43,126,339,946,2591,7254,20125,56450,158355,446618,
%T 1262225,3580686,10181479,29032254,82968843,237645250,682014587,
%U 1960981598,5647919645,16292761730,47069104613,136166703562,394418199725,1143822046786,3320790074371
%N Number of n-bead necklace structures using exactly three different colored beads.
%C Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.
%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 E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%F a(n) = A002076(n) - A000013(n).
%F From _Robert A. Russell_, May 29 2018: (Start)
%F a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 6] * (S2(n/d + 2, 3) - S2(n/d + 1, 3)) + [d==3 mod 6] * (S2(n/d + 2, 3) - 3*S2(n/d + 1, 3) + 3*S2(n/d, 3)) + [d==2 mod 6 | d==4 mod 6] * (2*S2(n/d + 1, 3) - 2*S2(n/d, 3)) + [d==1 mod 6 | d=5 mod 6] * S2(n/d, 3)), where S2(n,k) is the Stirling subset number, A008277.
%F G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 6] * (log(1-3x^d) - log(1-x^d)) + [d==3 mod 6] * (log(1-3x^d) - log(1-2x^d) + log(1-x^d)) / 2 + [d==2 mod 6 | d==4 mod 6] * (2*log(1-3x^d) - 3*log(1-2x^d)) / 3 + [d==1 mod 6 | d=5 mod 6] * (log(1-3x^d) - 3*log(1-2x^d) + 3*log(1-x^d)) / 6).
%F (End)
%t Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &], Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];
%t Table[Coefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/n , x, 3], {n, 1, 40}] (* _Robert A. Russell_, Feb 23 2018 *)
%t Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#,6], StirlingS2[n/#+2,3] - StirlingS2[n/#+1,3], Divisible[#,3], StirlingS2[n/#+2,3] - 3 StirlingS2[n/#+1,3] + 3 StirlingS2[n/#,3], Divisible[#,2], 2 StirlingS2[n/#+1,3] - 2 StirlingS2[n/#,3], True, StirlingS2[n/#,3]] &],{n, 1, 40}] (* _Robert A. Russell_, May 29 2018*)
%t mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[ Divisible[d, 6], Log[1 - 3x^d] - Log[1 - 2x^d], Divisible[d, 3] , (Log[1 - 3x^d] - Log[1 - 2x^d] + Log[1 - x^d]) / 2, Divisible[d, 2], (2 Log[1 - 3x^d] - 3 Log[1 - 2x^d]) / 3, True, (Log[1 - 3x^d] - 3Log[1 - 2x^d] + 3 Log[1 - x^d]) / 6], {d, 1, mx}], {x, 0, mx}], x], 1] (* _Robert A. Russell_, May 29 2018 *)
%Y Column 3 of A152175.
%Y Cf. A000013, A002076, A056283.
%K nonn
%O 1,4
%A _Marks R. Nester_