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%I #20 Feb 28 2019 19:57:56
%S 0,0,1,4,20,86,400,1852,8868,42892,210346,1038034,5150110,25623486,
%T 127740880,637539592,3184224728,15910524632,79520923966,397508610454,
%U 1987255480650,9935410066186,49674450471460,248364429410332,1241798688445588,6208922948527572,31044403310614786
%N Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).
%C Two color patterns are equivalent if the colors are permuted.
%C A chiral row is not equivalent to its reverse.
%C There are nonrecursive formulas, generating functions, and computer programs for A056272 and A305751, which can be used in conjunction with the first formula.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (11,-34,-16,247,-317,-200,610,-300).
%F a(n) = (A056272(n) - A305751(n))/2.
%F a(n) = A056272(n) - A056324(n).
%F a(n) = A056324(n) - A305751(n).
%F a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n).
%F 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)).
%F 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
%e For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.
%t LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {0, 0, 1, 4, 20, 86, 400, 1852}, 40] (* or *)
%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[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]
%Y Column 5 of A320751.
%Y Cf. A056272 (oriented), A056324 (unoriented), A305751 (achiral).
%K nonn,easy
%O 1,4
%A _Robert A. Russell_, Oct 27 2018