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%I #18 Nov 04 2019 02:17:41
%S 0,0,0,0,0,0,4,51,339,2010,10900,56700,286888,1426542,7014746,
%T 34229050,166197824,804243285,3883608940,18729354638,90266471623,
%U 434946282498,2096010533584,10104160993993,48733654211358,235195966291020,1135892493220025,5490005931157446,26555178320890184,128550000630553133,622790399873432344,3019641804537586657
%N Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 5 colors (subsets).
%C Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
%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 A056298 and A304975, which can be used in conjunction with the first formula.
%H Andrew Howroyd, <a href="/A320645/b320645.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) = (A056298(n) - A304975(n)) / 2 = A056298(n) - A056360(n) = A056360(n) - A304975(n).
%F 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 For a(7)=4, the chiral pairs are AABACDE-AABCDAE, AABCBDE-AABCDED, AABCDBE-AABCDEC, and ABACBDE-ABACDBE.
%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 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 k=5; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]
%Y Column 5 of A320647.
%Y Cf. A056298 (oriented), A056360 (unoriented), A304975 (achiral).
%K nonn,easy
%O 1,7
%A _Robert A. Russell_, Oct 18 2018
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