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%I #15 Nov 04 2019 12:45:42
%S 0,0,0,0,0,4,13,46,144,420,1221,3474,9856,27794,78632,222156,629760,
%T 1787440,5087797,14509580,41479867,118811286,341009901,980488510,
%U 2824029648,8146494860,23534997912,68084154502,197211336576,571915188840,1660405181149,4825559508106,14038010213051,40875403561680,119122661856133,347441159864556,1014152747485696
%N Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).
%C Two color patterns are equivalent if the colors are permuted.
%C Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%C There are nonrecursive formulas, generating functions, and computer programs for A002076 and A182522, which can be used in conjunction with the first formula.
%H Andrew Howroyd, <a href="/A320743/b320743.txt">Table of n, a(n) for n = 1..200</a>
%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) = (A002076(n) - A182522(n)) / 2 = A002076(n) - A056353(n) = A056353(n) - A182522(n).
%F a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=3 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
%F a(n) = A059053(n) + A320643(n).
%e For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.
%t Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
%t Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
%t k=3; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]
%Y Column 3 of A320742.
%Y Cf. A002076 (oriented), A056353 (unoriented), A182522 (achiral).
%K nonn,easy
%O 1,6
%A _Robert A. Russell_, Oct 21 2018
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