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 A331278 #15 Jan 23 2020 16:44:31
%S A331278 1,1,1,0,1,1,0,1,2,1,0,1,12,4,1,0,1,124,124,8,1,0,1,1800,10596,1280,
%T A331278 16,1,0,1,33648,1764244,930880,13456,32,1,0,1,769336,484423460,
%U A331278 1849386640,85835216,143808,64,1,0,1,20796960,198461691404,7798297361808,2098356708016,8206486848,1556416,128,1
%N A331278 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.
%C A331278 The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
%C A331278 A(n,k) is the number of n-uniform k-block sets of multisets.
%H A331278 Andrew Howroyd, <a href="/A331278/b331278.txt">Table of n, a(n) for n = 0..1325</a>
%F A331278 A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
%F A331278 A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A316674(n, j)/k!.
%F A331278 A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331315(n, j).
%F A331278 A(n, k) = A011782(n) * A331277(n, k) for k > 0.
%F A331278 A331395(n) = Sum_{d|n} A(n/d, d).
%e A331278 Array begins:
%e A331278 ====================================================================
%e A331278 n\k | 0 1 2 3 4 5
%e A331278 ----+---------------------------------------------------------------
%e A331278 0 | 1 1 0 0 0 0 ...
%e A331278 1 | 1 1 1 1 1 1 ...
%e A331278 2 | 1 2 12 124 1800 33648 ...
%e A331278 3 | 1 4 124 10596 1764244 484423460 ...
%e A331278 4 | 1 8 1280 930880 1849386640 7798297361808 ...
%e A331278 5 | 1 16 13456 85835216 2098356708016 140094551934813712 ...
%e A331278 6 | 1 32 143808 8206486848 2516779512105152 ...
%e A331278 ...
%e A331278 The A(2,2) matrices are:
%e A331278 [1 0] [1 0] [1 0] [2 0] [1 1] [1 0]
%e A331278 [1 0] [0 1] [0 1] [0 1] [1 0] [1 1]
%e A331278 [0 1] [1 0] [0 1] [0 1] [0 1] [0 1]
%e A331278 [0 1] [0 1] [1 0]
%e A331278 .
%e A331278 [1 0] [1 0] [1 0] [2 1] [2 0] [1 0]
%e A331278 [1 0] [0 2] [0 1] [0 1] [0 2] [1 2]
%e A331278 [0 2] [1 0] [1 1]
%o A331278 (PARI) T(n,k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1,n),k)*sum(i=j, m, (-1)^(i-j)*binomial(i,j)))}
%Y A331278 Rows n=1..2 are A000012, A173219.
%Y A331278 Columns k=0..2 are A000012, A011782, A331396.
%Y A331278 The version with binary entries is A331277.
%Y A331278 The version with not necessarily distinct columns is A331315.
%Y A331278 Cf. A316674 (unrestricted version), A330942, A331395.
%K A331278 nonn,tabl
%O A331278 0,9
%A A331278 _Andrew Howroyd_, Jan 13 2020