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 A320935 Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets). 4
 0, 0, 1, 4, 20, 86, 400, 1852, 8868, 42892, 210346, 1038034, 5150110, 25623486, 127740880, 637539592, 3184224728, 15910524632, 79520923966, 397508610454, 1987255480650, 9935410066186, 49674450471460, 248364429410332, 1241798688445588, 6208922948527572, 31044403310614786 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Two color patterns are equivalent if the colors are permuted. A chiral row is not equivalent to its reverse. There are nonrecursive formulas, generating functions, and computer programs for A056272 and A305751, which can be used in conjunction with the first formula. LINKS Index entries for linear recurrences with constant coefficients, signature (11,-34,-16,247,-317,-200,610,-300). FORMULA a(n) = (A056272(n) - A305751(n))/2. a(n) = A056272(n) - A056324(n). a(n) = A056324(n) - A305751(n). a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n). a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=5 is the maximum number of colors, S2 is the Stirling subset number A008277, and 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)). G.f.: x^3*(1 - 7*x + 10*x^2 + 18*x^3 - 49*x^4 + 25*x^5)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 5*x)*(1 - 5*x^2)*(1 - 2*x^2)). - Bruno Berselli, Oct 31 2018 EXAMPLE For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. MATHEMATICA LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {0, 0, 1, 4, 20, 86, 400, 1852}, 40] (* or *) 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[StirlingS2[n, j]-Ach[n, j], {j, k}]/2, {n, 40}] CROSSREFS Column 5 of A320751. Cf. A056272 (oriented), A056324 (unoriented), A305751 (achiral). Sequence in context: A262768 A250003 A343361 * A320936 A320937 A196953 Adjacent sequences:  A320932 A320933 A320934 * A320936 A320937 A320938 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 27 2018 STATUS approved

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Last modified May 18 19:20 EDT 2021. Contains 344001 sequences. (Running on oeis4.)