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 A331160 #11 Jan 24 2020 15:53:39
%S A331160 1,1,1,0,1,1,0,1,1,1,0,1,4,2,1,0,1,27,15,2,1,0,1,266,317,44,3,1,0,1,
%T A331160 3599,12586,2763,120,4,1,0,1,62941,803764,390399,21006,319,5,1,0,1,
%U A331160 1372117,75603729,103678954,10074052,147296,804,6,1
%N A331160 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 rows in decreasing lexicographic order.
%C A331160 The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.
%H A331160 Andrew Howroyd, <a href="/A331160/b331160.txt">Table of n, a(n) for n = 0..209</a>
%F A331160 A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219585(n, j).
%F A331160 A331318(n) = Sum_{d|n} A(n/d, d).
%e A331160 Array begins:
%e A331160 ===================================================================
%e A331160 n\k | 0 1 2 3 4 5 6
%e A331160 ----+--------------------------------------------------------------
%e A331160 0 | 1 1 0 0 0 0 0 ...
%e A331160 1 | 1 1 1 1 1 1 1 ...
%e A331160 2 | 1 1 4 27 266 3599 62941 ...
%e A331160 3 | 1 2 15 317 12586 803764 75603729 ...
%e A331160 4 | 1 2 44 2763 390399 103678954 46278915417 ...
%e A331160 5 | 1 3 120 21006 10074052 10679934500 21806685647346 ...
%e A331160 6 | 1 4 319 147296 232165926 956594630508 8717423133548684 ...
%e A331160 7 | 1 5 804 967829 4903530137 76812482919237 ...
%e A331160 ...
%e A331160 The A(2,2) = 4 matrices are:
%e A331160 [2 1] [2 0] [1 2] [1 1]
%e A331160 [0 1] [0 2] [1 0] [1 0]
%e A331160 [0 1]
%o A331160 (PARI)
%o A331160 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A331160 D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
%o A331160 T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}
%Y A331160 Rows n=1..3 are A000012, A331316, A331344
%Y A331160 Columns k=0..2 are A000012, A000009, A331317.
%Y A331160 Cf. A219585, A331039, A331126, A331161, A331315, A331318.
%K A331160 nonn,tabl
%O A331160 0,13
%A A331160 _Andrew Howroyd_, Jan 10 2020