The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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 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) = (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: A226375 A048995 A215639 * A328931 A235325 A230401 Adjacent sequences:  A320642 A320643 A320644 * A320646 A320647 A320648 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 18 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified February 23 21:20 EST 2020. Contains 332195 sequences. (Running on oeis4.)