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 A320646 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 6 colors (subsets). 3
 0, 0, 0, 0, 0, 0, 0, 9, 125, 1054, 7928, 54383, 356594, 2259504, 14008733, 85422360, 514773336, 3074341497, 18238301412, 107649939612, 632987843336, 3711471738408, 21716706883190, 126879832615600, 740528154956264, 4319137675225128, 25181504728152534, 146788320134425736, 855660631677225738, 4988501691655508510, 29089896998939710698 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse. 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 A056299 and A304976, 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) = (A056299(n) - A304976(n)) / 2 = A056299(n) - A056361(n) = A056361(n) - A304976(n). 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)). EXAMPLE 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. MATHEMATICA 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 *) 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]] k=6; Table[DivisorSum[n, EulerPhi[#]Adnk[#, n/#, k]&]/(2n) - Ach[n, k]/2, {n, 40}] CROSSREFS Column 6 of A320647. Cf. A056299 (oriented), A056361 (unoriented), A304976 (achiral). Sequence in context: A320529 A280896 A138438 * A241709 A085528 A192724 Adjacent sequences:  A320643 A320644 A320645 * A320647 A320648 A320649 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 19 2018 STATUS approved

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Last modified September 20 17:42 EDT 2021. Contains 347588 sequences. (Running on oeis4.)