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%I #25 Jul 28 2018 11:41:02
%S 0,0,0,0,0,1,3,36,296,2303,16317,110462,717024,4532105,28046285,
%T 170938814,1029749994,6149327905,36477979041,215304158916,
%U 1265984738264,7422971231829,43433472086235,253759842223290
%N Number of n-bead necklace structures using exactly six 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) = A056294(n) - A056293(n).
%F From _Robert A. Russell_, May 29 2018: (Start)
%F a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 60] * (S2(n/d+5,6) -
%F 10*S2(n/d+4,6) + 35*S2(n/d+3,6) - 50*S2(n/d+2,6) + 24*S2(n/d+1,6)) +
%F [d==30 mod 60] * (S2(n/d+5,6) - 12*S2(n/d+4,6) + 56*S2(n/d+3,6) -
%F 123*S2(n/d+2,6) + 108*S2(n/d+1,6)) + [d==20 mod 60 | d==40 mod 60] *
%F (4*S2(n/d+4,6) - 44*S2(n/d+3,6) + 176*S2(n/d+2,6) - 296*S2(n/d+1,6) +
%F 160*S2(n/d,6)) + [d==15 mod 60 | d==45 mod 60] * (3*S2(n/d+4,6) -
%F 36*S2(n/d+3,6) + 159*S2(n/d+2,6) - 306*S2(n/d+1,6) + 225*S2(n/d,6)) +
%F [d mod 60 in {12,24,36,48}] * (S2(n/d+5,6) - 12*S2(n/d+4,6) +
%F 59*S2(n/d+3,6) - 156*S2(n/d+2,6) + 228*S2(n/d+1,6) - 144*S2(n/d,6)) +
%F [d=10 mod 60 | d==50 mod 60] * (2*S2(n/d+4,6) - 23*S2(n/d+3,6) +
%F 103*S2(n/d+2,6) - 212*S2(n/d+1,6) + 160*S2(n/d,6)) + [d mod 60 in
%F {6,18,42,54}] * (S2(n/d+5,6) - 14*S2(n/d+4,6) + 80*S2(n/d+3,6) -
%F 229*S2(n/d+2,6) + 312*S2(n/d+1,6) - 144*S2(n/d,6)) + [d mod 60 in
%F {5,25,35,55}] * (2*S2(n/d+4,6) - 24*S2(n/d+3,6) + 106*S2(n/d+2,6) -
%F 204*S2(n/d+1,6) + 145*S2(n/d,6)) + [d mod 60 in {4,8,16,28,32,44,52,56}] *
%F (2*S2(n/d+4,6) - 20*S2(n/d+3,6) + 70*S2(n/d+2,6) - 92*S2(n/d+1,6) +
%F 16*S2(n/d,6)) + [d mod 60 in {3,9,21,27,33,39,51,57}] * (S2(n/d+4,6) -
%F 12*S2(n/d+3,6) + 53*S2(n/d+2,6) - 102*S2(n/d+1,6) + 81*S2(n/d,6)) +
%F [d mod 60 in {2,14,22,26,34,38,46,58}] * (S2(n/d+3,6) - 3*S2(n/d+2,6) -
%F 8*S2(n/d+1,6) + 16*S2(n/d,6)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,
%F 41,43,47,49,53,59}] * S2(n/d,6)), where S2(n,k) is the Stirling subset
%F number, A008277.
%F G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * (log(1-6x^d) -
%F log(1-5x^d)) + [d==30 mod 60] * (3*log(1-6x^d) - 3*log(1-5x^d) +
%F log(1-2x^d) - log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] *
%F (5*log(1-6x^d) - 6*log(1-5x^d) + 2*log(1-3x^d) - 3*log(1-2x^d)) / 9 +
%F [d==15 mod 60 | d==45 mod 60] * (5*log(1-6x^d) - 6*log(1-5x^d) +
%F 3*log(1-4x^d) - 4*log(1-3x^d) + 3*log(1-2x^d) - 6*log(1-x^d)) / 16 +
%F [d mod 60 in {12,24,36,48}] * (4*log(1-6x^d) - 4*log(1-5x^d) +
%F log(1-x^d)) / 5 + [d=10 mod 60 | d==50 mod 60] * (11*log(1-6x^d) -
%F 15*log(1-5x^d) + 8*log(1-3x^d) - 3*log(1-2x^d) - 9*log(1-x^d)) / 36 +
%F [d mod 60 in {6,18,42,54}] * (11*log(1-6x^d) - 11*log(1-5x^d) +
%F 5*log(1-2x^d) - log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] *
%F (29*log(1-6x^d) - 30*log(1-5x^d) + 3*log(1-4x^d) - 4*log(1-3x^d) +
%F 3*log(1-2x^d) - 30*log(1-x^d)) / 144 + [d mod 60 in {4,8,16,28,32,44,52,
%F 56}] * (16*log(1-6x^d) - 21*log(1-5x^d) + 10*log(1-3x^d) -
%F 15*log(1-2x^d) + 9*log(1-x^d)) / 45 + [d mod 60 in {3,9,21,27,33,39,51, 57}] * (9*log(1-6x^d) - 14*log(1-5x^d) + 15*log(1-4x^d) - 20*log(1-3x^d) +
%F 15*log(1-2x^d) - 14*log(1-x^d)) / 80 + [d mod 60 in {2,14,22,26,34,38,46,
%F 58}] * (19*log(1-6x^d) - 39*log(1-5x^d) + 40*log(1-3x^d) -
%F 15*log(1-2x^d) - 9*log(1-x^d)) / 180 + [d mod 60 in {1,7,11,13,17,19,23, 29,31,37,41,43,47,49,53,59}] * (log(1-6x^d) - 6 log(1-5x^d) +
%F 15 log(1-4x^d) - 20 log(1-3x^d) + 15 log(1-2x^d) - 6 log(1-x^d)) / 720).
%F (End)
%t From _Robert A. Russell_, May 29 2018: (Start)
%t Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &],
%t 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, 6],
%t {n, 1, 40}] (* after Gilbert and Riordan *)
%t Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], StirlingS2[n/#+5, 6] - 10 StirlingS2[n/#+4, 6] + 35 StirlingS2[n/#+3, 6] - 50 StirlingS2[n/#+2, 6] + 24 StirlingS2[n/#+1, 6], Divisible[#, 30], StirlingS2[n/#+5, 6] - 12 StirlingS2[n/#+4, 6] + 56 StirlingS2[n/#+3, 6] - 123 StirlingS2[n/#+2, 6] + 108 StirlingS2[n/#+1, 6], Divisible[#, 20], 4 StirlingS2[n/#+4, 6] - 44 StirlingS2[n/#+3, 6] + 176 StirlingS2[n/#+2, 6] - 296 StirlingS2[n/#+1, 6] + 160 StirlingS2[n/#, 6], Divisible[#, 15], 3 StirlingS2[n/#+4, 6] - 36 StirlingS2[n/#+3, 6] + 159 StirlingS2[n/#+2, 6] - 306 StirlingS2[n/#+1, 6] + 225 StirlingS2[n/#, 6], Divisible[#, 12], StirlingS2[n/#+5, 6] - 12 StirlingS2[n/#+4, 6] + 59 StirlingS2[n/#+3, 6] - 156 StirlingS2[n/#+2, 6] + 228 StirlingS2[n/#+1, 6] - 144 StirlingS2[n/#, 6], Divisible[#, 10], 2 StirlingS2[n/#+4, 6] - 23 StirlingS2[n/#+3, 6] + 103 StirlingS2[n/#+2, 6] - 212 StirlingS2[n/#+1, 6] + 160 StirlingS2[n/#, 6], Divisible[#, 6], StirlingS2[n/#+5, 6] - 14 StirlingS2[n/#+4, 6] + 80 StirlingS2[n/#+3, 6] - 229 StirlingS2[n/#+2, 6] + 312 StirlingS2[n/#+1, 6] - 144 StirlingS2[n/#, 6], Divisible[#, 5], 2 StirlingS2[n/#+4, 6] - 24 StirlingS2[n/#+3, 6] + 106 StirlingS2[n/#+2, 6] - 204 StirlingS2[n/#+1, 6] + 145 StirlingS2[n/#, 6], Divisible[#, 4], 2 StirlingS2[n/#+4, 6] - 20 StirlingS2[n/#+3, 6] + 70 StirlingS2[n/#+2, 6] - 92 StirlingS2[n/#+1, 6] + 16 StirlingS2[n/#, 6], Divisible[#, 3], StirlingS2[n/#+4, 6] - 12 StirlingS2[n/#+3, 6] + 53 StirlingS2[n/#+2, 6] - 102 StirlingS2[n/#+1, 6] + 81 StirlingS2[n/#, 6], Divisible[#, 2], StirlingS2[n/#+3, 6] - 3 StirlingS2[n/#+2, 6] - 8 StirlingS2[n/#+1, 6] + 16 StirlingS2[n/#, 6], True, StirlingS2[n/#, 6]] &], {n, 1, 40}]
%t mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[
%t Divisible[d, 60], Log[1 - 6x^d] - Log[1 - 5x^d], Divisible[d, 30],
%t (3 Log[1 - 6x^d] - 3 Log[1 - 5x^d] + Log[1 - 2x^d] - Log[1 - x^d]) / 4,
%t Divisible[d, 20], (5 Log[1 - 6x^d] - 6 Log[1 - 5x^d] + 2 Log[1 - 3x^d] -
%t 3 Log[1 - 2x^d]) / 9, Divisible[d, 15], (5 Log[1 - 6x^d] -
%t 6 Log[1 - 5x^d] + 3 Log[1 - 4x^d] - 4 Log[1 - 3x^d] + 3 Log[1 - 2x^d] -
%t 6 Log[1 - x^d]) / 16, Divisible[d, 12], (4 Log[1 - 6x^d] -
%t 4 Log[1 - 5x^d] + Log[1 - x^d]) / 5, Divisible[d, 10], (11 Log[1 - 6x^d] -
%t 15 Log[1 - 5x^d] + 8 Log[1 - 3x^d] - 3 Log[1 - 2x^d] - 9 Log[1 - x^d]) /
%t 36, Divisible[d, 6], (11 Log[1 - 6x^d] - 11 Log[1 - 5x^d] +
%t 5 Log[1 - 2x^d] - Log[1 - x^d]) / 20, Divisible[d, 5], (29 Log[1 - 6x^d] -
%t 30 Log[1 - 5x^d] + 3 Log[1 - 4x^d] - 4 Log[1 - 3x^d] + 3 Log[1 - 2x^d] -
%t 30 Log[1 - x^d]) / 144, Divisible[d, 4], (16 Log[1 - 6x^d] -
%t 21 Log[1 - 5x^d] + 10 Log[1 - 3x^d] - 15 Log[1 - 2x^d] + 9 Log[1 - x^d]) /
%t 45, Divisible[d, 3], (9 Log[1 - 6x^d] - 14 Log[1 - 5x^d] +
%t 15 Log[1 - 4x^d] - 20 Log[1 - 3x^d] + 15 Log[1 - 2x^d] -
%t 14 Log[1 - x^d]) / 80, Divisible[d, 2], (19 Log[1 - 6x^d] -
%t 39 Log[1 - 5x^d] + 40 Log[1 - 3x^d] - 15 Log[1 - 2x^d] - 9 Log[1 - x^d]) /
%t 180, True, (Log[1 - 6x^d] - 6 Log[1 - 5x^d] + 15 Log[1 - 4x^d] -
%t 20 Log[1 - 3x^d] + 15 Log[1 - 2x^d] - 6 Log[1 - x^d]) / 720],
%t {d, 1, mx}], {x, 0, mx}], x], 1]
%t (End)
%Y Column 6 of A152175.
%Y Cf. A056286, A056293, A056294.
%K nonn
%O 1,7
%A _Marks R. Nester_