This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320746 #13 Nov 04 2019 14:52:41
%S A320746 0,0,0,0,0,6,34,190,996,5070,26454,139484,749742,4082481,22509626,
%T A320746 125231540,702004040,3958071545,22423227634,127524417922,727617119592,
%U A320746 4163076477731,23876455868772,137228326265794,790200053665362,4557942281943078,26331297198477874,152331940294133402,882422871962784662,5117852332008063806,29715786649820358328,172720006045619486686,1004904748993330281274,5852047136464153694312
%N A320746 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets).
%C A320746 Two color patterns are equivalent if the colors are permuted.
%C A320746 Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%C A320746 There are nonrecursive formulas, generating functions, and computer programs for A056294 and A305752, which can be used in conjunction with the first formula.
%H A320746 Andrew Howroyd, <a href="/A320746/b320746.txt">Table of n, a(n) for n = 1..200</a>
%H A320746 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 A320746 a(n) = (A056294(n) - A305752(n)) / 2 = A056294(n) - A056356(n) = A056356(n) - A305752(n).
%F A320746 a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=6 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 A320746 a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n) + A320646(n).
%e A320746 For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.
%t A320746 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 A320746 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 A320746 k=6; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]
%Y A320746 Column 6 of A320742.
%Y A320746 Cf. A056294 (oriented), A056356 (unoriented), A305752 (achiral).
%K A320746 nonn,easy
%O A320746 1,6
%A A320746 _Robert A. Russell_, Oct 21 2018