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 A320526 #19 Mar 13 2024 13:37:24
%S A320526 0,0,0,2,10,40,141,464,1480,4600,14145,43052,130480,393820,1186521,
%T A320526 3568784,10725760,32213200,96714465,290284052,871142800,2613981700,
%U A320526 7843080201,23531425304,70598731840,211804847800,635432109585,1906330676252,5719061512720,17157321139180
%N A320526 a(n) is the number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 3 colors (subsets).
%C A320526 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 A320526 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6, -6, -24, 49, 6, -66, 36).
%F A320526 a(n) = (S2(n,k) - A(n,k))/2, where k=3 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 A320526 G.f.: (x^3 / Product_{k=1..3} (1 - k*x) - x^3*(1 + 2 x)/((1 - 2 x^2)*(1 - 3 x^2))) / 2.
%F A320526 a(n) = (A000392(n) - A304973(n)) / 2 = A000392(n) - A056327(n) = A056327(n) - A304973(n).
%e A320526 For a(4)=2, the two chiral pairs are AABC-ABCC and ABAC-ABCB.
%t A320526 k=3; Table[(StirlingS2[n,k] - If[EvenQ[n], 2StirlingS2[n/2+1,3] - 2StirlingS2[n/2,3], StirlingS2[(n+3)/2,3] - StirlingS2[(n+1)/2,3]])/2, {n, 1, 30}]
%t A320526 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 A320526 k = 3; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
%t A320526 LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 0, 2, 10, 40,
%t A320526 141}, 40]
%o A320526 (PARI) m=40; v=concat([0,0,0,2,10,40,141], vector(m-7)); for(n=8, m, v[n] = 6*v[n-1] -6*v[n-2] -24*v[n-3] +49*v[n-4] +6*v[n-5] -66*v[n-6] +36*v[n-7] ); v \\ _G. C. Greubel_, Oct 16 2018
%o A320526 (Magma) I:=[0,0,0,2,10,40,141]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -24*Self(n-3) +49*Self(n-4) +6*Self(n-5) -66*Self(n-6) +36*Self(n-7): n in [1..40]]; // _G. C. Greubel_, Oct 16 2018
%Y A320526 Column 3 of A320525.
%Y A320526 Cf. A000392 (oriented), A056327 (unoriented), A304973 (achiral).
%K A320526 nonn,easy
%O A320526 1,4
%A A320526 _Robert A. Russell_, Oct 14 2018