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 A320525 #26 Nov 05 2019 05:59:47
%S A320525 0,0,0,0,1,0,0,2,2,0,0,6,10,4,0,0,12,40,28,6,0,0,28,141,167,64,9,0,0,
%T A320525 56,464,824,508,124,12,0,0,120,1480,3840,3428,1300,220,16,0,0,240,
%U A320525 4600,16920,21132,11316,2900,360,20,0,0,496,14145,72655,123050,89513,31846,5890,560,25,0,0,992,43052,305140,688850,660978,313190,79256,11060,830,30,0
%N A320525 Triangle read by rows: T(n,k) = number of chiral pairs of color patterns (set partitions) in a row of length n using exactly k colors (subsets).
%C A320525 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.
%C A320525 If the top entry of the triangle is changed from 0 to 1, this is the number of non-equivalent distinguishing partitions of the path on n vertices (n >= 1) with exactly k parts (1 <= k <= n). - _Bahman Ahmadi_, Aug 21 2019
%H A320525 Andrew Howroyd, <a href="/A320525/b320525.txt">Table of n, a(n) for n = 1..1275</a>
%H A320525 B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, <a href="https://arxiv.org/abs/1910.12102">Number of Distinguishing Colorings and Partitions</a>, arXiv:1910.12102 [math.CO], 2019.
%F A320525 T(n,k) = (S2(n,k) - A(n,k))/2, where 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 A320525 T(n,k) = (A008277(n,k) - A304972(n,k)) / 2 = A008277(n,k) - A284949(n,k) = A284949(n,k) - A304972(n,k).
%e A320525 Triangle begins with T(1,1):
%e A320525 0;
%e A320525 0, 0;
%e A320525 0, 1, 0;
%e A320525 0, 2, 2, 0;
%e A320525 0, 6, 10, 4, 0;
%e A320525 0, 12, 40, 28, 6, 0;
%e A320525 0, 28, 141, 167, 64, 9, 0;
%e A320525 0, 56, 464, 824, 508, 124, 12, 0;
%e A320525 0, 120, 1480, 3840, 3428, 1300, 220, 16, 0;
%e A320525 0, 240, 4600, 16920, 21132, 11316, 2900, 360, 20, 0;
%e A320525 0, 496, 14145, 72655, 123050, 89513, 31846, 5890, 560, 25, 0;
%e A320525 0, 992, 43052, 305140, 688850, 660978, 313190, 79256, 11060, 830, 30, 0;
%e A320525 ...
%e A320525 For T(3,2)=1, the chiral pair is AAB-ABB. For T(4,2)=2, the chiral pairs are AAAB-ABBB and AABA-ABAA. For T(5,2)=6, the chiral pairs are AAAAB-ABBBB, AAABA-ABAAA, AAABB-AABBB, AABAB-ABABB, AABBA-ABBAA, and ABAAB-ABBAB.
%t A320525 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 A320525 Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 12}, {k, 1, n}] // Flatten
%o A320525 (PARI) \\ here Ach is A304972 as square matrix.
%o A320525 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 A320525 T(n)={(matrix(n,n,i,k,stirling(i,k,2)) - Ach(n))/2}
%o A320525 { my(A=T(10)); for(n=1, #A, print(A[n,1..n])) } \\ _Andrew Howroyd_, Sep 18 2019
%Y A320525 Columns 1-6 are A000004, A122746(n-2), A320526, A320527, A320528, A320529.
%Y A320525 Row sums are A320937.
%Y A320525 Cf. A008277 (oriented), A284949 (unoriented), A304972 (achiral).
%K A320525 nonn,tabl,easy
%O A320525 1,8
%A A320525 _Robert A. Russell_, Oct 14 2018