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%I #31 Dec 05 2022 08:14:30
%S 1,1,1,1,1,1,1,2,2,1,1,5,9,3,1,1,15,66,31,5,1,1,52,712,686,109,7,1,1,
%T 203,10457,27036,6721,339,11,1,1,877,198091,1688360,911838,58616,1043,
%U 15,1,1,4140,4659138,154703688,231575143,26908756,476781,2998,22,1
%N Number A(n,k) of k-partite partitions of {n}^k into k-tuples; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A(n,k) is the number of factorizations of m^n where m is a product of k distinct primes. A(2,2) = 9: (2*3)^2 = 36 has 9 factorizations: 36, 3*12, 4*9, 3*3*4, 2*18, 6*6, 2*3*6, 2*2*9, 2*2*3*3.
%C A(n,k) is the number of (n*k) X k matrices with nonnegative integer entries and column sums n up to permutation of rows. - _Andrew Howroyd_, Dec 10 2018
%H Andrew Howroyd, <a href="/A219727/b219727.txt">Table of n, a(n) for n = 0..209</a>
%e A(1,3) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(1,0,0),(0,1,0),(0,0,1)], [(0,1,1),(1,0,0)].
%e A(2,2) = 9: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(2,0),(0,1),(0,1)], [(1,2),(1,0)], [(1,1),(1,1)], [(1,1),(1,0),(0,1)], [(1,0),(1,0),(0,2)], [(1,0),(1,0),(0,1),(0,1)].
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 2, 5, 15, 52, 203, 877, ...
%e 1, 2, 9, 66, 712, 10457, 198091, 4659138, ...
%e 1, 3, 31, 686, 27036, 1688360, 154703688, ...
%e 1, 5, 109, 6721, 911838, 231575143, ...
%e 1, 7, 339, 58616, 26908756, ...
%e 1, 11, 1043, 476781, ...
%e 1, 15, 2998, ...
%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]]++); EulerT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
%o T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, 1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p,n,k), [1,n]); s*q[#q-j]))} \\ _Andrew Howroyd_, Dec 11 2018
%Y Columns k=0..3 give: A000012, A000041, A002774, A219678.
%Y Rows n=0..4 give: A000012, A000110, A020555, A322487, A358781.
%Y Main diagonal gives A322488.
%Y Cf. A188392, A219585 (partitions of {n}^k into distinct k-tuples), A256384, A318284, A318951.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Nov 26 2012
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