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 A320742 #14 Nov 04 2019 02:17:25
%S A320742 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,
%T A320742 1,0,0,0,0,0,0,6,13,2,0,0,0,0,0,0,6,30,46,7,0,0,0,0,0,0,6,34,130,144,
%U A320742 12,0,0,0,0,0,0,6,34,181,532,420,31,0,0,0,0,0,0,6,34,190,871,2006,1221,58,0,0,0,0,0,0,6,34,190,996,4016,7626,3474,126,0,0,0,0,0,0,6,34,190,1011,5070,18526,28401,9856,234,0
%N A320742 Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a cycle of length n using k or fewer colors (subsets).
%C A320742 Two color patterns are equivalent if the colors are permuted.
%C A320742 Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%H A320742 Andrew Howroyd, <a href="/A320742/b320742.txt">Table of n, a(n) for n = 1..1275</a>
%H A320742 E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%F A320742 T(n,k) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where 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)) and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
%F A320742 T(n,k) = (A320747(n,k) - A305749(n,k)) / 2 = A320747(n,k) - A320748(n,k)= A320748(n,k) - A305749(n,k).
%e A320742 Array begins with T(1,1):
%e A320742 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e A320742 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e A320742 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e A320742 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e A320742 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e A320742 0 0 4 6 6 6 6 6 6 6 6 6 ...
%e A320742 0 1 13 30 34 34 34 34 34 34 34 34 ...
%e A320742 0 2 46 130 181 190 190 190 190 190 190 190 ...
%e A320742 0 7 144 532 871 996 1011 1011 1011 1011 1011 1011 ...
%e A320742 0 12 420 2006 4016 5070 5328 5352 5352 5352 5352 5352 ...
%e A320742 0 31 1221 7626 18526 26454 29215 29705 29740 29740 29740 29740 ...
%e A320742 0 58 3474 28401 85101 139484 165164 171556 172415 172466 172466 172466 ...
%e A320742 For T(6,4)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD and AABCBD-AABCDC.
%t A320742 Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
%t A320742 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 A320742 Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j,k-n+1}], {k,15}, {n,k}] // Flatten
%o A320742 (PARI) \\ Ach is A304972 and R is A152175 as square matrices.
%o A320742 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 A320742 R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
%o A320742 T(n)={my(M=(R(n) - Ach(n))/2); for(i=2, n, M[,i] += M[,i-1]); M}
%o A320742 { my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ _Andrew Howroyd_, Nov 03 2019
%Y A320742 Partial row sums of A320647.
%Y A320742 Columns 1-6 are A000004, A059053, A320743, A320744, A320745, A320746
%Y A320742 For increasing k, columns converge to A320749.
%Y A320742 Cf. A320747 (oriented), A320748 (unoriented), A305749 (achiral).
%K A320742 nonn,tabl,easy
%O A320742 1,34
%A A320742 _Robert A. Russell_, Oct 21 2018