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 A320747 #13 Nov 04 2019 18:46:59
%S A320747 1,1,1,1,2,1,1,2,2,1,1,2,3,4,1,1,2,3,6,4,1,1,2,3,7,9,8,1,1,2,3,7,11,
%T A320747 26,10,1,1,2,3,7,12,39,53,20,1,1,2,3,7,12,42,103,146,30,1,1,2,3,7,12,
%U A320747 43,123,367,369,56,1,1,2,3,7,12,43,126,503,1235,1002,94,1,1,2,3,7,12,43,127,539,2008,4439,2685,180,1,1,2,3,7,12,43,127,543,2304,8720,15935,7434,316,1,1,2,3,7,12,43,127,544,2356,11023,38365,58509,20441,596,1
%N A320747 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an oriented cycle of length n using k or fewer colors (subsets).
%C A320747 Two color patterns are equivalent if the colors are permuted. An oriented cycle counts each chiral pair as two.
%C A320747 Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%C A320747 In other words, the number of n-bead necklace structures using a maximum of k different colored beads. - _Andrew Howroyd_, Oct 30 2019
%D A320747 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%H A320747 Andrew Howroyd, <a href="/A320747/b320747.txt">Table of n, a(n) for n = 1..1275</a>
%H A320747 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 A320747 T(n,k) = (1/n)*Sum_{j=1..k} Sum_{d|n} phi(d)*A(d,n/d,j), where 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 A320747 T(n,k) = A320748(n,k) + A320742(n,k) = 2*A320748(n,k) - A305749(n,k) = A305749(n,k) + 2*A320742(n,k).
%e A320747 Array begins with T(1,1):
%e A320747 1 1 1 1 1 1 1 1 1 1 1 1 ...
%e A320747 1 2 2 2 2 2 2 2 2 2 2 2 ...
%e A320747 1 2 3 3 3 3 3 3 3 3 3 3 ...
%e A320747 1 4 6 7 7 7 7 7 7 7 7 7 ...
%e A320747 1 4 9 11 12 12 12 12 12 12 12 12 ...
%e A320747 1 8 26 39 42 43 43 43 43 43 43 43 ...
%e A320747 1 10 53 103 123 126 127 127 127 127 127 127 ...
%e A320747 1 20 146 367 503 539 543 544 544 544 544 544 ...
%e A320747 1 30 369 1235 2008 2304 2356 2360 2361 2361 2361 2361 ...
%e A320747 1 56 1002 4439 8720 11023 11619 11697 11702 11703 11703 11703 ...
%e A320747 1 94 2685 15935 38365 54682 60499 61579 61684 61689 61690 61690 ...
%e A320747 1 180 7434 58509 173609 284071 336447 349746 351619 351766 351772 351773 ...
%e A320747 For T(4,2)=4, the patterns are AAAA, AAAB, AABB, and ABAB.
%e A320747 For T(4,3)=6, the patterns are the above four, AABC and ABAC.
%t A320747 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 A320747 Table[Sum[DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j] &], {j,k-n+1}]/n, {k,15}, {n,k}] // Flatten
%o A320747 (PARI) \\ R is A152175 as square matrix
%o A320747 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 A320747 T(n)={my(M=R(n)); for(i=2, n, M[,i] += M[,i-1]); M}
%o A320747 { my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ _Andrew Howroyd_, Nov 03 2019
%Y A320747 Partial row sums of A152175.
%Y A320747 Columns 1-6 are A057427, A000013, A002076, A056292, A056293, A056294.
%Y A320747 For increasing k, columns converge to A084423.
%Y A320747 Cf. A320748 (unoriented), A320742 (chiral), A305749 (achiral).
%K A320747 nonn,tabl,easy
%O A320747 1,5
%A A320747 _Robert A. Russell_, Oct 21 2018