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A320937 Number of chiral pairs of color patterns (set partitions) for a row of length n.

%I A320937 #12 Sep 19 2019 14:42:36
%S A320937 0,0,1,4,20,86,409,1988,10404,57488,338180,2103378,13814202,95423766,
%T A320937 691415451,5239857008,41431883216,341036489096,2916365967707,
%U A320937 25862060748614,237434856965694,2253357681164288,22076002386446896,222979432604192844,2319295160051570620
%N A320937 Number of chiral pairs of color patterns (set partitions) for a row of length n.
%C A320937 Two color patterns are equivalent if the colors are permuted.
%C A320937 A chiral row is not equivalent to its reverse.
%H A320937 Andrew Howroyd, <a href="/A320937/b320937.txt">Table of n, a(n) for n = 1..200</a>
%F A320937 a(n) = (A000110(n) + A080107(n)) / 2 = A000110(n) - A103293(n+1) = A103293(n+1) - A080107(n).
%F A320937 a(n) = Sum_{j=1..n} (S2(n,j) - Ach(n,j)) / 2, where 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)).
%e A320937 For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.
%t A320937 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 A320937 Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,n}]/2,{n,40}]
%o A320937 (PARI) \\ Ach is A304972 as square matrix.
%o A320937 Ach(n)={my(M=matrix(n,n,i,k,i>=k)); for(i=3, n, for(k=2, n, M[i,k]=k*M[i-2,k] + M[i-2,k-1] + if(k>2, M[i-2,k-2]))); M}
%o A320937 seq(n)={my(A=Ach(n)); vector(n, n, sum(k=1, n, stirling(n,k,2) - A[n,k])/2)} \\ _Andrew Howroyd_, Sep 18 2019
%Y A320937 Row sums of triangle A320525.
%Y A320937 Limit as k increases of column k of array A320751.
%Y A320937 Cf. A000110 (oriented), A103293 (unoriented), A080107 (achiral).
%K A320937 nonn,easy
%O A320937 1,4
%A A320937 _Robert A. Russell_, Oct 27 2018