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%I #26 Sep 08 2022 08:46:23
%S 0,0,0,0,0,6,64,508,3428,21132,123050,688850,3752350,20032446,
%T 105372624,548066568,2826316248,14478890712,73794322750,374602205590,
%U 1895629599050,9568906539786,48208435317284,242500368793628,1218342441784468,6115097961883092,30669103347259650,153720181809997530,770100204404335350,3856500105221902326
%N Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).
%C 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.
%H Colin Barker, <a href="/A320528/b320528.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (13,-48,-36,551,-683,-1542,3546,80,-4280,2400).
%F 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].
%F 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.
%F a(n) = (A000481(n) - A304975(n)) / 2 = A000481(n) - A056329(n) = A056329(n) - A304975(n).
%F 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
%e For a(6)=6, the chiral pairs are AABCDE-ABCDEE, ABACDE-ABCDED, ABCADE-ABCDEC, ABCDAE-ABCDEB, ABBCDE-ABCDDE, and ABCBDE-ABCDCE.
%t 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}]
%t 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 *)
%t k = 5; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
%t LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132}, 30]
%o (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
%o (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
%Y Col. 5 of A320525.
%Y Cf. A000481 (oriented), A056329 (unoriented), A304975 (achiral).
%K nonn,easy
%O 1,6
%A _Robert A. Russell_, Oct 14 2018
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