This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320529 #18 Sep 08 2022 08:46:23
%S A320529 0,0,0,0,0,0,9,124,1300,11316,89513,660978,4658738,31711638,210332227,
%T A320529 1367416232,8752773288,55343303064,346540112781,2153037307846,
%U A320529 13292835205606,81652655795106,499484899831775,3045117929546220,18513208314957356,112297592929814292,679900657841661529,4110073054119135194,24814158520762637754
%N A320529 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 6 colors (subsets).
%C A320529 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 A320529 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600).
%F A320529 a(n) = (S2(n,k) - A(n,k))/2, where k=6 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 A320529 G.f.: (x^6 / (Product_{k=1..6} (1 - k*x)) - x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.
%F A320529 a(n) = (A000770(n) - A304976(n)) / 2 = A000770(n) - A056330(n) = A056330(n) - A304976(n).
%e A320529 For a(7)=9, the chiral pairs are AABCDEF-ABCDEFF, ABACDEF-ABCDEFE, ABCADEF-ABCDEFD, ABCDAEF-ABCDEFC, ABCDEAF-ABCDEFB, ABBCDEF-ABCDEEF, ABCBDEF-ABCDEDF, ABCDBEF-ABCDECF, and ABCCDEF-ABCDDEF.
%t A320529 k=6; Table[(StirlingS2[n,k] - If[EvenQ[n], StirlingS2[n/2+3,6] - 3StirlingS2[n/2+2,6] - 8StirlingS2[n/2+1,6] + 16StirlingS2[n/2,6], 3StirlingS2[(n+5)/2,6] - 17StirlingS2[(n+3)/2,6] + 20StirlingS2[(n+1)/2,6]])/2, {n,30}]
%t A320529 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 A320529 k = 6; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
%t A320529 LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638}, 30]
%o A320529 (PARI) x='x+O('x^30); concat(vector(6), Vec((x^6/prod(k=1,6, 1-k*x) - x^6* (1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/prod(k=1,6,(1-k*x^2)))/2)) \\ _G. C. Greubel_, Oct 19 2018
%o A320529 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0,0,0,0,0,0] cat Coefficients(R!((x^6/(&*[1-k*x: k in [1..6]]) - x^6*(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/(&*[1-k*x^2: k in [1..6]]) )/2)); // _G. C. Greubel_, Oct 19 2018
%Y A320529 Column 6 of A320525.
%Y A320529 Cf. A000770 (oriented), A056330 (unoriented), A304976 (achiral).
%K A320529 nonn,easy
%O A320529 1,7
%A A320529 _Robert A. Russell_, Oct 14 2018