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A056297
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Number of n-bead necklace structures using exactly four different colored beads.
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7
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0, 0, 0, 1, 2, 13, 50, 221, 866, 3437, 13250, 51075, 194810, 742651, 2823766, 10738881, 40843370, 155494751, 592614050, 2261625725, 8643289534, 33080920607, 126797503250, 486710971595, 1870851589554, 7201014763285, 27752927359726, 107092397450897
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OFFSET
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1,5
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COMMENTS
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Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.
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REFERENCES
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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]
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LINKS
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FORMULA
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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.
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).
(End)
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MATHEMATICA
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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, 4], {n, 1, 40}] (* after Gilbert and Riordan *)
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}]
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]
(End)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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