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A056297
Number of n-bead necklace structures using exactly four different colored beads.
7
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
OFFSET
1,5
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) = A056292(n) - A002076(n).
From Robert A. Russell, May 29 2018: (Start)
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)
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, 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)
CROSSREFS
Column 4 of A152175.
Sequence in context: A319759 A254784 A056305 * A241892 A037383 A034476
KEYWORD
nonn
STATUS
approved