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%I #12 Jan 23 2020 16:45:33
%S 1,1,1,0,1,1,0,1,1,1,0,1,6,1,1,0,1,62,31,1,1,0,1,900,2649,160,1,1,0,1,
%T 16824,441061,116360,841,1,1,0,1,384668,121105865,231173330,5364701,
%U 4494,1,1,0,1,10398480,49615422851,974787170226,131147294251,256452714,24319,1,1
%N Array read by antidiagonals: A(n,k) is the number of binary matrices with k distinct columns and any number of nonzero rows with n ones in every column and columns in decreasing lexicographic order.
%C The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
%C A(n,k) is the number of labeled n-uniform hypergraphs with k edges and no isolated vertices. When n=2 these objects are graphs.
%H Andrew Howroyd, <a href="/A331277/b331277.txt">Table of n, a(n) for n = 0..1325</a>
%F A(n, k) = Sum_{j=0..n*k} binomial(binomial(j,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
%F A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A262809(n, j)/k!.
%F A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A330942(n, j).
%F A331639(n) = Sum_{d|n} A(n/d, d).
%e Array begins:
%e ====================================================================
%e n\k | 0 1 2 3 4 5 6
%e ----+---------------------------------------------------------------
%e 0 | 1 1 0 0 0 0 0 ...
%e 1 | 1 1 1 1 1 1 1 ...
%e 2 | 1 1 6 62 900 16824 384668 ...
%e 3 | 1 1 31 2649 441061 121105865 49615422851 ...
%e 4 | 1 1 160 116360 231173330 974787170226 ...
%e 5 | 1 1 841 5364701 131147294251 ...
%e 6 | 1 1 4494 256452714 78649359753286 ...
%e ...
%e The A(2,2) = 6 matrices are:
%e [1 0] [1 0] [1 0] [1 1] [1 0] [1 0]
%e [1 0] [0 1] [0 1] [1 0] [1 1] [0 1]
%e [0 1] [1 0] [0 1] [0 1] [0 1] [1 1]
%e [0 1] [0 1] [1 0]
%o (PARI) T(n,k)={my(m=n*k); sum(j=0, m, binomial(binomial(j,n), k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}
%Y Rows n=1..3 are A000012, A121251, A136245.
%Y Columns k=0..3 are A000012, A000012, A047665, A137219.
%Y The version with nonnegative integer entries is A331278.
%Y The version with not necessarily distinct columns is A330942.
%Y Cf. A262809 (unrestricted version), A331315, A331639.
%K nonn,tabl
%O 0,13
%A _Andrew Howroyd_, Jan 13 2020