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 A056298 Number of n-bead necklace structures using exactly five different colored beads. 5
 0, 0, 0, 0, 1, 3, 20, 136, 773, 4281, 22430, 115100, 577577, 2863227, 14051164, 68515514, 332514803, 1608800691, 7767857090, 37460388596, 180536313547, 869901397479, 4192038616700, 20208367895980 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure. REFERENCES 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] LINKS E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. FORMULA a(n) = A056293(n) - A056292(n). From Robert A. Russell, May 29 2018: (Start) a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 60] * (S2(n/d+4,5) -   6*S2(n/d+3,5) + 11*S2(n/d+2,5) - 6*S2(n/d+1,5)) + [d==30 mod 60] *   (S2(n/d+4,5) - 8*S2(n/d+3,5) + 26*S2(n/d+2,5) - 43*S2(n/d+1,5) +   30*S2(n/d,5)) + [d==20 mod 60 | d==40 mod 60] * (S2(n/d+4,5) -   8*S2(n/d+3,5) + 23*S2(n/d+2,5) - 24*S2(n/d+1,5)) + [d==15 mod 60 |   d==45 mod 60] * (S2(n/d+4,5) - 10*S2(n/d+3,5) + 38*S2(n/d+2,5) -   65*S2(n/d+1,5) + 45*S2(n/d,5)) + [d mod 60 in {12,24,36,48}] *   (4*S2(n/d+3,5) - 24*S2(n/d+2,5) + 44*S2(n/d+1,5) - 24*S2(n/d,5)) +   [d=10 mod 60 | d==50 mod 60] * (S2(n/d+4,5) - 10*S2(n/d+3,5) +   38*S2(n/d+2,5) - 61*S2(n/d+1,5) + 30*S2(n/d,5)) + [d mod 60 in   {6,18,42,54}] * (2*S2(n/d+3,5) - 9*S2(n/d+2,5) + 7*S2(n/d+1,5) +   6*S2(n/d,5)) + [d mod 60 in {5,25,35,55}] * (S2(n/d+4,5) -   10*S2(n/d+3,5) + 35*S2(n/d+2,5) - 50*S2(n/d+1,5) + 25*S2(n/d,5)) +   [d mod 60 in {4,8,16,28,32,44,52,56}] * (2*S2(n/d+3,5) - 12*S2(n/d+2,5) +   26*S2(n/d+1,5) - 24*S2(n/d,5)) + [d mod 60 in {3,9,21,27,33,39,51,57}] *   (3*S2(n/d+2,5) - 15*S2(n/d+1,5) + 21*S2(n/d,5)) + [d mod 60 in   {2,14,22,26,34,38,46,58}] * (3*S2(n/d+2,5) - 11*S2(n/d+1,5) +   6*S2(n/d,5)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,   59}] * S2(n/d,5)), where S2(n,k) is the Stirling subset number, A008277. G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * (log(1-4x^d) -   log(1-3x^d)) + [d==30 mod 60] * (3*log[1-5x^d) - 3*log(1-4x^d) +   log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] * (2*log(1-5x^d) -   2*log(1-4x^d) + log(1-2x^d) - log(1-x^d)) / 3 +   [d==15 mod 60 | d==45 mod 60] * (3*log(1-5x^d) - 3*log(1-4x^d) +   2*log(1-3x^d) - 2*log(1-2x^d) + 3*log(1-x^d)) / 8 + [d mod 60 in   {12,24,36,48}] * (4*log(1-5x^d) - 5*log(1-4x^d)) / 5 + [d=10 mod 60 |   d==50 mod 60] * (5*log(1-5x^d) - 5*log(1-4x^d) + 4*log(1-2x^d) -   log(1-x^d)) / 12 + [d mod 60 in {6,18,42,54}] * (11*log(1-5x^d) -   15*log(1-4x^d) + 5*log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] *   (5*log(1-5x^d) - log(1-4x^d) + 2*log(1-3x^d) - 2*log(1-2x^d) +   log(1-x^d)) / 24 + [d mod 60 in {4,8,16,28,32,44,52,56}] *   (7*log(1-5x^d) - 10*log(1-4x^d) + 5*log(1-2x^d) - 5*log(1-x^d)) /   15 + [d mod 60 in {3,9,21,27,33,39,51,57}] * (7*log(1-5x^d) -   15*log(1-4x^d) + 10*log(1-3x^d) - 10*log(1-2x^d) + 15*log(1-x^d)) /   40 + [d mod 60 in {2,14,22,26,34,38,46,58}] * (13*log(1-5x^d) -   25*log(1-4x^d) + 20*log(1-2x^d) - 5*log(1-x^d)) / 60 + [d mod 60 in   {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}] * (log(1-5x^d) -   5*log(1-4x^d) + 10*log(1-3x^d) - 10*log(1-2x^d) + 5*log(1-x^d)) / 120). (End) MATHEMATICA From Robert A. Russell, May 29 2018: (Start) 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]]; Table[Coefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/n , x, 5],   {n, 1, 40}] (* after Gilbert and Riordan *) Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], StirlingS2[n/#+4, 5] - 6 StirlingS2[n/#+3, 5] + 11 StirlingS2[n/#+2, 5] - 6 StirlingS2[n/#+1, 5], Divisible[#, 30], StirlingS2[n/#+4, 5] - 8 StirlingS2[n/#+3, 5] + 26 StirlingS2[n/#+2, 5] - 43 StirlingS2[n/#+1, 5] + 30 StirlingS2[n/#, 5], Divisible[#, 20], StirlingS2[n/#+4, 5] - 8 StirlingS2[n/#+3, 5] + 23 StirlingS2[n/#+2, 5] - 24 StirlingS2[n/#+1, 5], Divisible[#, 15], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 38 StirlingS2[n/#+2, 5] - 65 StirlingS2[n/#+1, 5] + 45 StirlingS2[n/#, 5], Divisible[#, 12], 4 StirlingS2[n/#+3, 5] - 24 StirlingS2[n/#+2, 5] + 44 StirlingS2[n/#+1, 5] - 24 StirlingS2[n/#, 5], Divisible[#, 10], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 38 StirlingS2[n/#+2, 5] - 61 StirlingS2[n/#+1, 5] + 30 StirlingS2[n/#, 5], Divisible[#, 6], 2 StirlingS2[n/#+3, 5] - 9 StirlingS2[n/#+2, 5] + 7 StirlingS2[n/#+1, 5] + 6 StirlingS2[n/#, 5], Divisible[#, 5], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 35 StirlingS2[n/#+2, 5] - 50 StirlingS2[n/#+1, 5] + 25 StirlingS2[n/#, 5], Divisible[#, 4], 2 StirlingS2[n/#+3, 5] - 12 StirlingS2[n/#+2, 5] + 26 StirlingS2[n/#+1, 5] - 24 StirlingS2[n/#, 5], Divisible[#, 3], 3 StirlingS2[n/#+2, 5] - 15 StirlingS2[n/#+1, 5] + 21 StirlingS2[n/#, 5], Divisible[#, 2], 3 StirlingS2[n/#+2, 5] - 11 StirlingS2[n/#+1, 5] + 6 StirlingS2[n/#, 5], True, StirlingS2[n/#, 5]] &], {n, 1, 40}] mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[   Divisible[d, 60], Log[1 - 5x^d] - Log[1 - 4x^d], Divisible[d, 30],   (3 Log[1 - 5x^d] - 3 Log[1 - 4x^d] + Log[1 - x^d]) / 4, Divisible[d, 20],   (2 Log[1 - 5x^d] - 2 Log[1 - 4x^d] + Log[1 - 2x^d] - Log[1 - x^d]) / 3,   Divisible[d, 15], (3 Log[1 - 5x^d] - 3 Log[1 - 4x^d] + 2 Log[1 - 3x^d] -   2 Log[1 - 2x^d] + 3 Log[1 - x^d]) / 8, Divisible[d, 12],   (4 Log[1 - 5x^d] - 5 Log[1 - 4x^d]) / 5, Divisible[d, 10],   (5 Log[1 - 5x^d] - 5 Log[1 - 4x^d] + 4 Log[1 - 2x^d] - Log[1 - x^d]) / 12,   Divisible[d, 6], (11 Log[1 - 5x^d] - 15 Log[1 - 4x^d] + 5 Log[1 - x^d]) /   20, Divisible[d, 5], (5 Log[1 - 5x^d] - Log[1 - 4x^d] + 2 Log[1 - 3x^d] -   2 Log[1 - 2x^d] + Log[1 - x^d]) / 24, Divisible[d, 4], (7 Log[1 - 5x^d] -   10 Log[1 - 4x^d] + 5 Log[1 - 2x^d] - 5 Log[1 - x^d]) / 15,   Divisible[d, 3], (7 Log[1 - 5x^d] - 15 Log[1 - 4x^d] + 10 Log[1 - 3x^d] -   10 Log[1 - 2x^d] + 15 Log[1 - x^d]) / 40, Divisible[d, 2],   (13 Log[1 - 5x^d] - 25 Log[1 - 4x^d] + 20 Log[1 - 2x^d] -   5 Log[1 - x^d]) / 60, True, (Log[1 - 5x^d] - 5 Log[1 - 4x^d] +   10 Log[1 - 3x^d] - 10 Log[1 - 2x^d] + 5 Log[1 - x^d]) / 120], {d, 1, mx}], {x, 0, mx}], x], 1] (End) CROSSREFS Column 5 of A152175. Cf. A056285, A056292, A056293. Sequence in context: A000276 A216778 A056306 * A114479 A074574 A267899 Adjacent sequences:  A056295 A056296 A056297 * A056299 A056300 A056301 KEYWORD nonn AUTHOR STATUS approved

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Last modified March 28 17:51 EDT 2020. Contains 333103 sequences. (Running on oeis4.)