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A056293 Number of n-bead necklace structures using a maximum of five different colored beads. 9
1, 2, 3, 7, 12, 42, 123, 503, 2008, 8720, 38365, 173609, 792828, 3662924, 17034381, 79703081, 374624254, 1767883444, 8370666417, 39751072847, 189262621864, 903220058756, 4319518316899, 20697040198889, 99343899144822, 477609477924308, 2299585449279713 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Vincenzo Librandi, Table of n, a(n) for n = 1..500

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

N. J. A. Sloane, Maple code for this and related sequences

FORMULA

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

From Robert A. Russell, May 29 2018: (Start)

a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 60] * (5*S2(n/d + 4, 5) - 50*S2(n/d + 3, 5) + 175*S2(n/d + 2, 5) - 250*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d==30 mod 60] * (4*S2(n/d+4,5) - 41*S2(n/d+3,5) + 149*S2(n/d+2,5) - 226*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d==20 mod 60 | d==40 mod 60] * (4*S2(n/d + 4, 5) - 42*S2(n/d + 3, 5) + 156*S2(n/d + 2, 5) - 238*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d==15 mod 60 | d==45 mod 60] * (3*S2(n/d + 4, 5) - 33*S2(n/d + 3, 5) + 129*S2(n/d + 2, 5) - 210*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d mod 60 in {12,24,36,48}] * (4*S2(n/d + 4, 5) - 40*S2(n/d + 3, 5) + 140*S2(n/d + 2, 5) - 200*S2(n/d+1, 5) + 96*S2(n/d, 5)) + [d=10 mod 60 | d==50 mod 60] * (3*S2(n/d + 4, 5) - 33*S2(n/d + 3, 5) + 130*S2(n/d + 2, 5) - 214*S2(n/d + 1, 5) + 120*S2(n/d, 5)) + [d mod 60 in {6,18,42,54}] * (3*S2(n/d + 4, 5) - 31*S2(n/d + 3, 5) + 114*S2(n/d + 2, 5) - 176*S2(n/d + 1, 5) + 96*S2(n/d, 5)) + [d mod 60 in {5,25,35,55}] * (2*S2(n/d + 4, 5) - 23*S2(n/d + 3, 5) + 95*S2(n/d + 2, 5) - 165*S2(n/d + 1, 5) + 100*S2(n/d, 5)) + [d mod 60 in {4,8,16,28,32,44,52,56}] * (3*S2(n/d + 4, 5) - 32*S2(n/d + 3, 5) + 121*S2(n/d + 2, 5) - 188*S2(n/d + 1, 5) + 96*S2(n/d, 5)) + [d mod 60 in {3,9,21,27,33,39,51,57}] * (2*S2(n/d + 4, 5) - 23*S2(n/d + 3, 5) + 94*S2(n/d + 2, 5) - 160*S2(n/d + 1, 5) + 96*S2(n/d, 5)) + [d mod 60 in {2,14,22,26,34,38,46,58}] * (2*S2(n/d + 4, 5) - 23*S2(n/d + 3, 5) + 95*S2(n/d + 2, 5) - 164*S2(n/d + 1, 5) + 96*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 + 4, 5) - 13*S2(n/d + 3, 5) + 60*S2(n/d + 2, 5) - 115*S2(n/d + 1, 5) + 76*S2(n/d, 5))), where S2(n,k) is the Stirling subset number, A008277.

G.f.: 1 - Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * log(1-5x^d) + [d==30 mod 60] * (3*log(1-5x^d) + log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] * (2*log(1-5x^d) + log(1-2x^d)) / 3 + [d==15 mod 60 | d==45 mod 60] * (3*log(1-5x^d) + 2*log(1-3x^d) + 3*log(1-x^d)) / 8 + [d mod 60 in {12,24,36,48}] * 4*log(1-5x^d) / 5 + [d=10 mod 60 | d==50 mod 60] * (5*log(1-5x^d) + 4*log(1-2x^d) + 3*log(1-x^d)) / 12 + [d mod 60 in {6,18,42,54}] * (11*log(1-5x^d) + 5*log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] * (5*log(1-5x^d) + 2*log(1-3x^d) + 4*log(1-2x^d) + 9*log(1-x^d)) / 24 + [d mod 60 in {4,8,16,28,32,44,52,56}] * (7*log(1-5x^d) + 5*log(1-2x^d)) / 15 + [d mod 60 in {3,9,21,27,33,39,51,57}] * (7*log(1-5x^d) + 10*log(1-3x^d) + 15*log(1-x^d)) / 40 + [d mod 60 in {2,14,22,26,34,38,46,58}] * (13*log(1-5x^d) + 20*log(1-2x^d) + 15*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) + 10*log(1-3x^d) + 20*log(1-2x^d) + 45*log(1-x^d)) / 120).

(End)

MATHEMATICA

Adn[d_, n_] := Module[{ c, t1, t2}, t2 = 0; For[c = 1, c <= d, c++, If[Mod[d, c] == 0 , t2 = t2 + (x^c/c)*(E^(c*z) - 1)]]; t1 = E^t2; t1 = Series[t1, {z, 0, n+1}]; Coefficient[t1, z, n]*n!]; Pn[n_] := Module[{ d, e, t1}, t1 = 0; For[d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*Adn[d, n/d]/n]]; t1/(1 - x)]; Pnq[n_, q_] := Module[{t1}, t1 = Series[Pn[n], {x, 0, q+1}] ; Coefficient[t1, x, q]]; a[n_] := Pnq[n, 5]; Table[Print[an = a[n]]; an, {n, 1, 24}] (* Jean-Fran├žois Alcover, Oct 04 2013, after N. J. A. Sloane's Maple code *)

(* this Mathematica program uses Gilbert and Riordan's recurrence formula, which they recommend for calculations: *)

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[SeriesCoefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]

/(n (1 - x)), {x, 0, 5}], {n, 1, 40}] (* Robert A. Russell, Feb 24 2018 *)

From Robert A. Russell, May 29 2018: (Start)

Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], 5 StirlingS2[n/#+4, 5] - 50 StirlingS2[n/#+3, 5] + 175 StirlingS2[n/#+2, 5] - 250 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 30], 4 StirlingS2[n/#+4, 5] - 41 StirlingS2[n/#+3, 5] + 149 StirlingS2[n/#+2, 5] - 226 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 20], 4 StirlingS2[n/#+4, 5] - 42 StirlingS2[n/#+3, 5] + 156 StirlingS2[n/#+2, 5] - 238 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 15], 3 StirlingS2[n/#+4, 5] - 33 StirlingS2[n/#+3, 5] + 129 StirlingS2[n/#+2, 5] - 210 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 12], 4 StirlingS2[n/#+4, 5] - 40 StirlingS2[n/#+3, 5] + 140 StirlingS2[n/#+2, 5] - 200 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], Divisible[#, 10], 3 StirlingS2[n/#+4, 5] - 33 StirlingS2[n/#+3, 5] + 130 StirlingS2[n/#+2, 5] - 214 StirlingS2[n/#+1, 5] + 120 StirlingS2[n/#, 5], Divisible[#, 6], 3 StirlingS2[n/#+4, 5] - 31 StirlingS2[n/#+3, 5] + 114 StirlingS2[n/#+2, 5] - 176 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], Divisible[#, 5], 2 StirlingS2[n/#+4, 5] - 23 StirlingS2[n/#+3, 5] + 95 StirlingS2[n/#+2, 5] - 165 StirlingS2[n/#+1, 5] + 100 StirlingS2[n/#, 5], Divisible[#, 4], 3 StirlingS2[n/#+4, 5] - 32 StirlingS2[n/#+3, 5] + 121 StirlingS2[n/#+2, 5] - 188 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], Divisible[#, 3], 2 StirlingS2[n/#+4, 5] - 23 StirlingS2[n/#+3, 5] + 94 StirlingS2[n/#+2, 5] - 160 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], Divisible[#, 2], 2 StirlingS2[n/#+4, 5] - 23 StirlingS2[n/#+3, 5] + 95 StirlingS2[n/#+2, 5] - 164 StirlingS2[n/#+1, 5] + 96 StirlingS2[n/#, 5], True, StirlingS2[n/#+4, 5] - 13 StirlingS2[n/#+3, 5] + 60 StirlingS2[n/#+2, 5] - 115 StirlingS2[n/#+1, 5] + 76 StirlingS2[n/#, 5]] &], {n, 1, 40}]

mx = 40; Drop[CoefficientList[Series[1-Sum[(EulerPhi[d] / d) Which[ Divisible[d, 60], Log[1-5x^d], Divisible[d, 30], (3 Log[1-5x^d] + Log[1-x^d]) / 4, Divisible[d, 20], (2 Log[1-5x^d] + Log[1-2x^d]) / 3, Divisible[d, 15], (3 Log[1-5x^d] + 2 Log[1-3x^d] + 3 Log[1-x^d]) / 8, Divisible[d, 12], 4 Log[1-5x^d] / 5, Divisible[d, 10], (5 Log[1-5x^d] + 4 Log[1-2x^d] + 3 Log[1-x^d]) / 12, Divisible[d, 6], (11 Log[1-5x^d] + 5 Log[1-x^d]) / 20, Divisible[d, 5], (5 Log[1-5x^d] + 2 Log[1-3x^d] + 4 Log[1-2x^d] + 9 Log[1-x^d]) / 24, Divisible[d, 4], (7 Log[1-5x^d] + 5 Log[1-2x^d]) / 15, Divisible[d, 3], (7 Log[1-5x^d] + 10 Log[1-3x^d] + 15 Log[1-x^d]) / 40, Divisible[d, 2], (13 Log[1-5x^d] + 20 Log[1-2x^d] + 15 Log[1-x^d]) / 60, True, (Log[1-5x^d] + 10 Log[1-3x^d] + 20 Log[1-2x^d] + 45 Log[1-x^d]) / 120], {d, 1, mx}], {x, 0, mx}], x], 1]

(End)

CROSSREFS

Cf. A000013, A001869.

Sequence in context: A084708 A035003 A143879 * A275311 A056294 A084423

Adjacent sequences:  A056290 A056291 A056292 * A056294 A056295 A056296

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

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Last modified April 8 03:01 EDT 2020. Contains 333312 sequences. (Running on oeis4.)