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%I #7 Sep 20 2019 19:37:52
%S 0,0,0,0,0,6,34,190,1011,5352,29740,172466,1055232,6793791,46034940,
%T 327303819,2436650368,18944771253,153488081102,1293086505784,
%U 11306373089104,102425178180769,959825673145688,9290807818971900,92771800581171418,954447025978145744,10105871186441842623,110009631951698573068,1229996584263621368224,14112483571723367245825,166021918475962174194914,2001010469483653602192695
%N Number of chiral pairs of color patterns (set partitions) in a cycle of length n.
%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.
%H Andrew Howroyd, <a href="/A320749/b320749.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) = Sum_{j=1..n} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where 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) = (A084423(n) - A080107(n)) / 2 = A084423(n) - A084708(n) = A084708(n) - A080107(n).
%e For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.
%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]]
%t Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j,n}], {n,40}]
%Y Row sums of A320647.
%Y Columns of A320742 converge to this as k increases.
%Y Cf. A084423 (oriented), A084708 (unoriented), A080107 (achiral).
%K nonn,easy
%O 1,6
%A _Robert A. Russell_, Oct 22 2018
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