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A320645
Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 5 colors (subsets).
4
0, 0, 0, 0, 0, 0, 4, 51, 339, 2010, 10900, 56700, 286888, 1426542, 7014746, 34229050, 166197824, 804243285, 3883608940, 18729354638, 90266471623, 434946282498, 2096010533584, 10104160993993, 48733654211358, 235195966291020, 1135892493220025, 5490005931157446, 26555178320890184, 128550000630553133, 622790399873432344, 3019641804537586657
OFFSET
1,7
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 A056298 and A304975, which can be used in conjunction with the first formula.
LINKS
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
FORMULA
a(n) = (A056298(n) - A304975(n)) / 2 = A056298(n) - A056360(n) = A056360(n) - A304975(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(7)=4, the chiral pairs are AABACDE-AABCDAE, AABCBDE-AABCDED, AABCDBE-AABCDEC, and ABACBDE-ABACDBE.
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=5; Table[DivisorSum[n, EulerPhi[#]Adnk[#, n/#, k]&]/(2n) - Ach[n, k]/2, {n, 40}]
CROSSREFS
Column 5 of A320647.
Cf. A056298 (oriented), A056360 (unoriented), A304975 (achiral).
Sequence in context: A347551 A048995 A215639 * A347921 A328931 A343572
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 18 2018
STATUS
approved