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%I #12 Jan 26 2020 16:27:59
%S 1,1,1,1,1,1,1,2,1,1,1,4,7,3,1,1,8,59,45,3,1,1,16,701,1987,271,5,1,1,
%T 32,10460,190379,73567,1244,11,1,1,64,190816,30474159,58055460,
%U 2451082,7289,13,1,1,128,4098997,7287577611,100171963518,16557581754,75511809,40841,19,1
%N Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of distinct nonzero rows with column sums n and columns in nonincreasing lexicographic order.
%C The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
%H Andrew Howroyd, <a href="/A331572/b331572.txt">Table of n, a(n) for n = 0..209</a>
%F A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A331568(n, j)/k!.
%F A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331570(n, j).
%F A331713(n) = Sum_{d|n} A(n/d, d).
%e Array begins:
%e ==========================================================
%e n\k | 0 1 2 3 4 5
%e ----+-----------------------------------------------------
%e 0 | 1 1 1 1 1 1 ...
%e 1 | 1 1 2 4 8 16 ...
%e 2 | 1 1 7 59 701 10460 ...
%e 3 | 1 3 45 1987 190379 30474159 ...
%e 4 | 1 3 271 73567 58055460 100171963518 ...
%e 5 | 1 5 1244 2451082 16557581754 311419969572540 ...
%e 6 | 1 11 7289 75511809 4388702900099 ...
%e ...
%e The A(2,2) = 7 matrices are:
%e [1 1] [1 0] [1 0] [2 1] [2 0] [1 0] [2 2]
%e [1 0] [1 1] [0 1] [0 1] [0 2] [1 2]
%e [0 1] [0 1] [1 1]
%o (PARI)
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n] + k - 1, k)/prod(i=1, #v, i^v[i]*v[i]!)}
%o 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, 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 Rows n=0..3 are A000012, A011782, A331709, A331710.
%Y Columns k=0..3 are A000012, A032020, A331711, A331712.
%Y Cf. A331315, A331568, A331569, A331570, A331571, A331713.
%K nonn,tabl
%O 0,8
%A _Andrew Howroyd_, Jan 21 2020
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