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 A320745 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 5 or fewer colors (subsets). 3
 0, 0, 0, 0, 0, 6, 34, 181, 871, 4016, 18526, 85101, 393148, 1822977, 8500893, 39809180, 187230704, 883730048, 4184926222, 19874478310, 94629276256, 451604739323, 2159748985582, 10348493650194, 49671898709098, 238804606717950, 1149792470325340, 5543620159707666, 26762240285558924, 129350640352555296, 625889650880647630, 3031651402693863747, 14698911258326292182, 71332938143655936584, 346474231506471943759 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Two color patterns are equivalent if the colors are permuted. Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference. There are nonrecursive formulas, generating functions, and computer programs for A056293 and A305751, which can be used in conjunction with the first formula. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. FORMULA a(n) = (A056293(n) - A305751(n)) / 2 = A056293(n) - A056355(n) = A056355(n) - A305751(n). a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=5 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)). a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n). EXAMPLE For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC. MATHEMATICA 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]] 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 *) k=5; Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#, n/#, j]&]/n - Ach[n, j])/2, {j, k}], {n, 40}] CROSSREFS Column 5 of A320742. Cf. A056293 (oriented), A056355 (unoriented), A305751 (achiral). Sequence in context: A274405 A144142 A126643 * A084775 A327740 A229009 Adjacent sequences: A320742 A320743 A320744 * A320746 A320747 A320748 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 21 2018 STATUS approved

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Last modified December 6 21:00 EST 2022. Contains 358648 sequences. (Running on oeis4.)