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 A320527 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets). 5
 0, 0, 0, 0, 4, 28, 167, 824, 3840, 16920, 72655, 305140, 1265264, 5193188, 21173607, 85887984, 347150080, 1399355440, 5629755935, 22615859180, 90754215024, 363888497148, 1458169977847, 5840524999144, 23385639542720, 93613165023560, 374664497695215, 1499293455643620, 5999080285068784, 24002040333605908 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern. LINKS FORMULA a(n) = (S2(n,k) - A(n,k))/2, where k=4 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0]. G.f.: (x^4 / Product_{k=1..4} (1 - k*x) - x^4*(1 + x)^2*(1 - 2 x^2) / Product_{k=1..4} (1 - k*x^2)) / 2. a(n) = (A000453(n) - A304974(n)) / 2 = A000453(n) - A056328(n) = A056328(n) - A304974(n). EXAMPLE For a(5)=4, the chiral pairs are AABCD-ABCDD, ABACD-ABCDC, ABBCD-ABCCD and ABCAD-ABCDB. MATHEMATICA k=4; Table[(StirlingS2[n, k] - If[EvenQ[n], StirlingS2[n/2+2, 4] - StirlingS2[n/2+1, 4] - 2StirlingS2[n/2, 4], 2StirlingS2[(n+3)/2, 4] - 4StirlingS2[(n+1)/2, 4]])/2, {n, 30}] 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[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}] LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 0, 4, 28, 167}, 30] CROSSREFS Col. 4 of A320525. Cf. A000453 (oriented), A056328 (unoriented), A304974 (achiral). Sequence in context: A272936 A053524 A270943 * A272281 A182190 A317232 Adjacent sequences:  A320524 A320525 A320526 * A320528 A320529 A320530 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 14 2018 STATUS approved

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Last modified October 14 11:49 EDT 2019. Contains 327996 sequences. (Running on oeis4.)