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 A320646 #16 Nov 04 2019 02:17:35
%S A320646 0,0,0,0,0,0,0,9,125,1054,7928,54383,356594,2259504,14008733,85422360,
%T A320646 514773336,3074341497,18238301412,107649939612,632987843336,
%U A320646 3711471738408,21716706883190,126879832615600,740528154956264,4319137675225128,25181504728152534,146788320134425736,855660631677225738,4988501691655508510,29089896998939710698
%N A320646 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 6 colors (subsets).
%C A320646 Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
%C A320646 Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%C A320646 There are nonrecursive formulas, generating functions, and computer programs for A056299 and A304976, which can be used in conjunction with the first formula.
%H A320646 Andrew Howroyd, <a href="/A320646/b320646.txt">Table of n, a(n) for n = 1..200</a>
%H A320646 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 A320646 a(n) = (A056299(n) - A304976(n)) / 2 = A056299(n) - A056361(n) = A056361(n) - A304976(n).
%F A320646 a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=5 is number of colors or sets, 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)).
%e A320646 For a(8)=9, the chiral pairs are AABACDEF-AABCDEAF, AABCADEF-AABCDAEF, AABCBDEF-AABCDEFE, AABCDBEF-AABCDEFD, AABCDEBF-AABCDEFC, AABCDCEF-AABCDEDF, ABACDEBF-ABACDEBF, ABCADBEF-ABCADECF, and ABCDAEBF-ABCADBEF.
%t A320646 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 A320646 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 A320646 k=6; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]
%Y A320646 Column 6 of A320647.
%Y A320646 Cf. A056299 (oriented), A056361 (unoriented), A304976 (achiral).
%K A320646 nonn,easy
%O A320646 1,8
%A A320646 _Robert A. Russell_, Oct 19 2018