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 A331569 #10 Jan 25 2020 17:55:26
%S A331569 1,1,1,0,1,1,0,1,0,1,0,1,3,0,1,0,1,17,0,0,1,0,1,230,184,0,0,1,0,1,
%T A331569 3264,16936,840,0,0,1,0,1,60338,2711904,768785,0,0,0,1,0,1,1287062,
%U A331569 675457000,1493786233,21770070,0,0,0,1,0,1,31900620,232383728378,5254074934990,585810653616,328149360,0,0,0,1
%N A331569 Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of distinct nonzero rows with n ones in every column and columns in decreasing lexicographic order.
%C A331569 The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
%C A331569 A(n,k) is the number of k-block n-uniform T_0 set systems without isolated vertices.
%H A331569 Andrew Howroyd, <a href="/A331569/b331569.txt">Table of n, a(n) for n = 0..209</a>
%F A331569 A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A331567(n, j)/k!.
%F A331569 A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331571(n, j).
%F A331569 A331651(n) = Sum_{d|n} A(n/d, d).
%e A331569 Array begins:
%e A331569 ===============================================================
%e A331569 n\k | 0 1 2 3 4 5 6
%e A331569 ----+----------------------------------------------------------
%e A331569 0 | 1 1 0 0 0 0 0 ...
%e A331569 1 | 1 1 1 1 1 1 1 ...
%e A331569 2 | 1 0 3 17 230 3264 60338 ...
%e A331569 3 | 1 0 0 184 16936 2711904 675457000 ...
%e A331569 4 | 1 0 0 840 768785 1493786233 5254074934990 ...
%e A331569 5 | 1 0 0 0 21770070 585810653616 30604798810581906 ...
%e A331569 6 | 1 0 0 0 328149360 161087473081920 ...
%e A331569 ...
%e A331569 The A(2,2) = 3 matrices are:
%e A331569 [1 1] [1 0] [1 0]
%e A331569 [1 0] [1 1] [0 1]
%e A331569 [0 1] [0 1] [1 1]
%o A331569 (PARI)
%o A331569 WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o A331569 D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n], k)/prod(i=1, #v, i^v[i]*v[i]!)}
%o A331569 T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, k<=1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }
%Y A331569 Rows n=1..4 are A000012, A331649, A094631, A331650.
%Y A331569 Cf. A330942, A331039, A331567, A331570, A331571, A331572, A331651.
%K A331569 nonn,tabl
%O A331569 0,13
%A A331569 _Andrew Howroyd_, Jan 20 2020