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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.)