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 A331315 #10 Jan 23 2020 16:46:24
%S A331315 1,1,1,1,1,1,1,2,2,1,1,4,14,4,1,1,8,150,128,8,1,1,16,2210,10848,1288,
%T A331315 16,1,1,32,41642,1796408,933448,13472,32,1,1,64,956878,491544512,
%U A331315 1852183128,85862144,143840,64,1,1,128,25955630,200901557728,7805700498776,2098614254048,8206774496,1556480,128,1
%N A331315 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n and columns in nonincreasing lexicographic order.
%C A331315 The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
%C A331315 A(n,k) is the number of n-uniform k-block multisets of multisets.
%H A331315 Andrew Howroyd, <a href="/A331315/b331315.txt">Table of n, a(n) for n = 0..1325</a>
%F A331315 A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n)+k-1, k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
%F A331315 A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A316674(n, j)/k!.
%F A331315 A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331278(n, j).
%F A331315 A(n, k) = A011782(n) * A330942(n, k) for k > 0.
%F A331315 A317583(n) = Sum_{d|n} A(n/d, d).
%e A331315 Array begins:
%e A331315 ====================================================================
%e A331315 n\k | 0 1 2 3 4 5
%e A331315 ----+---------------------------------------------------------------
%e A331315 0 | 1 1 1 1 1 1 ...
%e A331315 1 | 1 1 2 4 8 16 ...
%e A331315 2 | 1 2 14 150 2210 41642 ...
%e A331315 3 | 1 4 128 10848 1796408 491544512 ...
%e A331315 4 | 1 8 1288 933448 1852183128 7805700498776 ...
%e A331315 5 | 1 16 13472 85862144 2098614254048 140102945876710912 ...
%e A331315 6 | 1 32 143840 8206774496 2516804131997152 ...
%e A331315 ...
%e A331315 The A(2,2) = 14 matrices are:
%e A331315 [1 0] [1 0] [1 0] [2 0] [1 1] [1 0] [1 0]
%e A331315 [1 0] [0 1] [0 1] [0 1] [1 0] [1 1] [1 0]
%e A331315 [0 1] [1 0] [0 1] [0 1] [0 1] [0 1] [0 2]
%e A331315 [0 1] [0 1] [1 0]
%e A331315 .
%e A331315 [1 0] [1 0] [2 1] [2 0] [1 1] [1 0] [2 2]
%e A331315 [0 2] [0 1] [0 1] [0 2] [1 1] [1 2]
%e A331315 [1 0] [1 1]
%o A331315 (PARI) T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1, n)+k-1, k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}
%Y A331315 Rows n=1..2 are A000012, A121227.
%Y A331315 Columns k=0..2 are A000012, A011782, A331397.
%Y A331315 The version with binary entries is A330942.
%Y A331315 The version with distinct columns is A331278.
%Y A331315 Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
%Y A331315 All solutions: A316674 (all), A331568 (distinct rows).
%Y A331315 Up to row permutation: A219727, A219585, A331161, A331160.
%Y A331315 Up to column permutation: this sequence, A331572, A331278, A331570.
%Y A331315 Nonisomorphic: A331485.
%Y A331315 Cf. A317583.
%K A331315 nonn,tabl
%O A331315 0,8
%A A331315 _Andrew Howroyd_, Jan 13 2020