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%I #32 Sep 04 2018 12:20:47
%S 0,0,0,1,2,13,50,221,866,3437,13250,51075,194810,742651,2823766,
%T 10738881,40843370,155494751,592614050,2261625725,8643289534,
%U 33080920607,126797503250,486710971595,1870851589554,7201014763285,27752927359726,107092397450897
%N Number of n-bead necklace structures using exactly four 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) = A056292(n) - A002076(n).
%F From _Robert A. Russell_, May 29 2018: (Start)
%F a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 12] * (S2(n/d + 3, 4) - 3*S2(n/d+2,4) + 2*S2(n/d + 1, 4)) + [d==6 mod 12] * (3*S2(n/d + 2, 4) - 9*S2(n/d + 1, 4) + 3*S2(n/d, 4)) + [d==4 mod 12 | d==8 mod 12] * (S2(n/d + 3, 4) - 5*S2(n/d + 2, 4) - 10*S2(n/d + 1, 4) - 8*S2(n/d, 4)) + [d==3 mod 12 | d=9 mod 12] * (2*S2(n/d + 2, 4) - 8*S2(n/d + 1, 4) - 2*S2(n/d,4)) + [d==2 mod 12 | d==10 mod 12] * (S2(n/d + 2, 4) - S2(n/d + 1, 4) + 9*S2(n/d, 4)) + [d mod 12 in {1,5,7,11}] * S2(n/d, 4)), where S2(n,k) is the Stirling subset number, A008277.
%F G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 12] * (log(1-4x^d) - log(1-3x^d)) +[d==6 mod 12] * (3*log(1-4x^d) - 4*log(1-3x^d)) / 4 + [d==4 mod 12 | d==8 mod 12] * (2*log(1-4x^d) - 2*log(1-3x^d) + log(1-x^d)) / 3 + [d==3 mod 12 | d==9 mod 12] * (3*log(1-4x^d) - 4*log(1-3x^d) + 2*log(1-2x^d) - 4*log(1-x^d)) / 8 + [d==2 mod 12 | d=10 mod 12] * (5*log(1-4x^d) - 8*log(1-3x^d) + 4*log(1-x^d)) / 12 + [d mod 12 in {1,5,7,11}] * (log(1-4x^d) - 4*log(1-3x^d) + 6*log(1-2x^d) - 4*log(1-x^d)) / 24).
%F (End)
%t From _Robert A. Russell_, May 29 2018: (Start)
%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, 4], {n, 1, 40}] (* after Gilbert and Riordan *)
%t Table[(1/n) DivisorSum[n, EulerPhi[#] Which[ Divisible[#,12], StirlingS2[n/#+3,4] - 3 StirlingS2[n/#+2,4] + 2 StirlingS2[n/#+1,4], Divisible[#,6], 3 StirlingS2[n/#+2,4] - 9 StirlingS2[n/#+1,4] + 6 StirlingS2[n/#,4], Divisible[#,4], StirlingS2[n/#+3,4] - 5 StirlingS2[n/#+2,4] + 10 StirlingS2[n/#+1,4] - 8 StirlingS2[n/#,4], Divisible[#,3], 2 StirlingS2[n/#+2,4] - 8 StirlingS2[n/#+1,4] + 9 StirlingS2[n/#,4], Divisible[#,2], StirlingS2[n/#+2,4] - StirlingS2[n/#+1,4] - 2 StirlingS2[n/#,4], True, StirlingS2[n/#,4]] &],{n, 1, 40}]
%t mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[ Divisible[d, 12], Log[1-4x^d] - Log[1-3x^d], Divisible[d, 6], (3 Log[1-4x^d] - 4 Log[1-3x^d]) / 4, Divisible[d, 4], (2 Log[1-4x^d] - 2 Log[1-3x^d] + Log[1-x^d]) / 3, Divisible[d, 3], (3 Log[1-4x^d] - 4 Log[1-3x^d] + 2 Log[1-2x^d] - 4 Log[1-x^d]) / 8, Divisible[d, 2], (5 Log[1-4x^d] - 8 Log[1-3x^d] + 4 Log[1-x^d]) / 12, True, (Log[1-4x^d] - 4 Log[1-3x^d] + 6 Log[1-2x^d] - 4 Log[1-x^d]) / 24], {d, 1, mx}], {x, 0, mx}], x], 1]
%t (End)
%Y Column 4 of A152175.
%Y Cf. A002076, A056284, A056292.
%K nonn
%O 1,5
%A _Marks R. Nester_
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