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%I #13 Nov 04 2019 14:53:02
%S 0,0,0,0,0,6,30,130,532,2006,7626,28401,106260,396435,1486147,5580130,
%T 21032880,79486763,301317282,1145123672,4362804633,16658456825,
%U 63738451998,244332656201,938244497740,3608640426930,13899977105315,53614228550220,207061964668740,800639722002163,3099251007215286,12009598156277090,46582685655751645,180850428684482360
%N Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 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 A056292 and A305750, which can be used in conjunction with the first formula.
%H Andrew Howroyd, <a href="/A320744/b320744.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) = (A056292(n) - A305750(n)) / 2 = A056292(n) - A056354(n) = A056354(n) - A305750(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=4 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) + A320644(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]] (* A304972 *)
%t k=4; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]
%Y Column 4 of A320742.
%Y Cf. A056292 (oriented), A056354 (unoriented), A305750 (achiral).
%K nonn,easy
%O 1,6
%A _Robert A. Russell_, Oct 21 2018
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