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 A320644 #19 Mar 26 2023 14:31:56
%S A320644 0,0,0,0,0,2,17,84,388,1586,6405,24927,96404,368641,1407515,5357974,
%T A320644 20403120,77699323,296229485,1130614092,4321324766,16539645539,
%U A320644 63397442097,243352167691,935420468092,3600493932070,13876442107403,53546144395718,206864753332164,800067806813323,3097590602034137,12004772596768984,46568647645538594,180809553280920680
%N A320644 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 4 colors (subsets).
%C A320644 Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
%C A320644 Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%C A320644 There are nonrecursive formulas, generating functions, and computer programs for A056297 and A304974, which can be used in conjunction with the first formula.
%H A320644 Andrew Howroyd, <a href="/A320644/b320644.txt">Table of n, a(n) for n = 1..200</a>
%H A320644 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 A320644 a(n) = (A056297(n) - A304974(n)) / 2 = A056297(n) - A056359(n) = A056359(n) - A304974(n).
%F A320644 a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=4 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 A320644 For a(6)=2, the chiral pairs are AABACD-AABCAD and AABCBD-AABCDC.
%t A320644 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 A320644 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 A320644 k=4; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]
%Y A320644 Column 4 of A320647.
%Y A320644 Cf. A056297 (oriented), A056359 (unoriented), A304974 (achiral).
%K A320644 nonn,easy
%O A320644 1,6
%A A320644 _Robert A. Russell_, Oct 18 2018