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 A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets). 3
 0, 0, 1, 4, 20, 80, 336, 1344, 5440, 21760, 87296, 349184, 1397760, 5591040, 22368256, 89473024, 357908480, 1431633920, 5726601216, 22906404864, 91625881600, 366503526400, 1466015154176, 5864060616704, 23456246661120, 93824986644480, 375299963355136, 1501199853420544, 6004799480791040 (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 A124303 and A305750, which can be used in conjunction with the first formula. LINKS Index entries for linear recurrences with constant coefficients, signature (4,4,-16). FORMULA a(n) = (A124303(n) - A305750(n))/2. a(n) = A124303(n) - A056323(n). a(n) = A056323(n) - A305750(n). a(n) = A122746(n-2) + A320526(n) + A320527(n). a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=4 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)). a(2*m) = (16^m - 4*4^m)/48. a(2*m-1) = (16^m - 4*4^m)/192. a(n) = (4^n - 4^floor(n/2+1))/48. G.f.: x^2/((-1 + 4*x)*(-1 + 4*x^2)). - Stefano Spezia, Oct 29 2018 a(n) = 2^n*(2^n - (-1)^n - 3)/48. - Bruno Berselli, Oct 31 2018 EXAMPLE For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. MATHEMATICA Table[(4^n - 4^Floor[n/2+1])/48, {n, 40}] (* or *) LinearRecurrence[{4, 4, -16}, {0, 0, 1}, 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=4; Table[Sum[StirlingS2[n, j]-Ach[n, j], {j, k}]/2, {n, 40}] CoefficientList[Series[x^2/((-1 + 4 x) (-1 + 4 x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *) CROSSREFS Column 4 of A320751. Cf. A124303 (oriented), A056323 (unoriented), A305750 (achiral). Sequence in context: A074358 A255050 A292540 * A055296 A140532 A217482 Adjacent sequences:  A320931 A320932 A320933 * A320935 A320936 A320937 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 27 2018 STATUS approved

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Last modified October 22 17:29 EDT 2019. Contains 328319 sequences. (Running on oeis4.)