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 A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets). 5
 0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132, 123050, 688850, 3752350, 20032446, 105372624, 548066568, 2826316248, 14478890712, 73794322750, 374602205590, 1895629599050, 9568906539786, 48208435317284, 242500368793628, 1218342441784468, 6115097961883092, 30669103347259650, 153720181809997530, 770100204404335350, 3856500105221902326 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 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 Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (13,-48,-36,551,-683,-1542,3546,80,-4280,2400). FORMULA a(n) = (S2(n,k) - A(n,k))/2, where k=5 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^5 / Product_{k=1..5} (1 - k*x) - x^5 (1 + x) (1 - 3 x^2) (1 + 2 x - 2 x^2) / Product_{k=1..5} (1 - k*x^2)) / 2. a(n) = (A000481(n) - A304975(n)) / 2 = A000481(n) - A056329(n) = A056329(n) - A304975(n). a(n) = 13*a(n-1) - 48*a(n-2) - 36*a(n-3) + 551*a(n-4) - 683*a(n-5) - 1542*a(n-6) + 3546*a(n-7) + 80*a(n-8) - 4280*a(n-9) + 2400*a(n-10) for n>10. - Colin Barker, May 12 2020 EXAMPLE For a(6)=6, the chiral pairs are AABCDE-ABCDEE, ABACDE-ABCDED, ABCADE-ABCDEC, ABCDAE-ABCDEB, ABBCDE-ABCDDE, and ABCBDE-ABCDCE. MATHEMATICA k=5; Table[(StirlingS2[n, k] - If[EvenQ[n], 3StirlingS2[n/2+2, 5] - 11StirlingS2[n/2+1, 5] + 6StirlingS2[n/2, 5], StirlingS2[(n+5)/2, 5] - 3StirlingS2[(n+3)/2, 5]])/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 = 5; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}] LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132}, 30] PROG (PARI) m=30; v=concat([0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132], vector(m-10)); for(n=11, m, v[n] = 13*v[n-1]-48*v[n-2]-36*v[n-3]+551*v[n-4]-683*v[n-5] -1542*v[n-6] +3546*v[n-7] +80*v[n-8] -4280*v[n-9] +2400*v[n-10]); v \\ G. C. Greubel, Oct 20 2018 (Magma) I:=[0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132]; [n le 10 select I[n] else 13*Self(n-1)-48*Self(n-2)-36*Self(n-3)+551*Self(n-4)-683*Self(n-5) -1542*Self(n-6)+3546*Self(n-7)+80*Self(n-8)-4280*Self(n-9) +2400*Self(n-10): n in [1..30]]; // G. C. Greubel, Oct 20 2018 CROSSREFS Col. 5 of A320525. Cf. A000481 (oriented), A056329 (unoriented), A304975 (achiral). Sequence in context: A222596 A067447 A083225 * A237357 A230282 A186668 Adjacent sequences: A320525 A320526 A320527 * A320529 A320530 A320531 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 14 2018 STATUS approved

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Last modified January 31 05:36 EST 2023. Contains 359947 sequences. (Running on oeis4.)