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 A331570 #11 Jan 26 2020 16:28:42
%S A331570 1,1,1,0,1,1,0,1,1,1,0,1,6,3,1,0,1,46,42,3,1,0,1,544,1900,268,5,1,0,1,
%T A331570 7983,184550,73028,1239,11,1,0,1,144970,29724388,57835569,2448599,
%U A331570 7278,13,1,0,1,3097825,7137090958,99940181999,16550232235,75497242,40828,19,1
%N A331570 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of distinct nonzero rows with column sums n and columns in decreasing lexicographic order.
%C A331570 The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
%H A331570 Andrew Howroyd, <a href="/A331570/b331570.txt">Table of n, a(n) for n = 0..209</a>
%F A331570 A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A331568(n, j)/k!.
%F A331570 A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331572(n, j).
%F A331570 A331708(n) = Sum_{d|n} A(n/d, d).
%e A331570 Array begins:
%e A331570 =============================================================
%e A331570 n\k | 0 1 2 3 4 5
%e A331570 ----+--------------------------------------------------------
%e A331570 0 | 1 1 0 0 0 0 ...
%e A331570 1 | 1 1 1 1 1 1 ...
%e A331570 2 | 1 1 6 46 544 7983 ...
%e A331570 3 | 1 3 42 1900 184550 29724388 ...
%e A331570 4 | 1 3 268 73028 57835569 99940181999 ...
%e A331570 5 | 1 5 1239 2448599 16550232235 311353753947045 ...
%e A331570 6 | 1 11 7278 75497242 4388476386528 896320470282357104 ...
%e A331570 ...
%e A331570 The A(2,2) = 6 matrices are:
%e A331570 [1 1] [1 0] [1 0] [2 1] [2 0] [1 0]
%e A331570 [1 0] [1 1] [0 1] [0 1] [0 2] [1 2]
%e A331570 [0 1] [0 1] [1 1]
%o A331570 (PARI)
%o A331570 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A331570 D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n], k)/prod(i=1, #v, i^v[i]*v[i]!)}
%o A331570 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 A331570 Rows 1..3 are A000012, A331704, A331705.
%Y A331570 Columns k=0..3 are A000012, A032020, A331706, A331707.
%Y A331570 Cf. A331315, A331568, A331569, A331571, A331572, A331708.
%K A331570 nonn,tabl
%O A331570 0,13
%A A331570 _Andrew Howroyd_, Jan 21 2020