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%I #33 Sep 16 2019 04:29:24
%S 1,1,1,1,1,1,1,2,1,1,1,5,5,2,1,1,15,40,17,2,1,1,52,457,364,46,3,1,1,
%T 203,6995,14595,2897,123,4,1,1,877,136771,937776,407287,21369,323,5,1,
%U 1,4140,3299218,88507276,107652681,10200931,148257,809,6,1
%N Number A(n,k) of k-partite partitions of {n}^k into distinct 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 into distinct factors where m is a product of k distinct primes. A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into distinct factors: 36, 3*12, 4*9, 2*18, 2*3*6.
%H Andrew Howroyd, <a href="/A219585/b219585.txt">Table of n, a(n) for n = 0..209</a>
%F A(n,k) = [(Product_{j=1..k} x_j)^n] 1/2 * Product_{i_1,...,i_k>=0} (1+Product_{j=1..k} x_j^i_j).
%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(3,2) = 17: [(3,3)], [(3,0),(0,3)], [(3,2),(0,1)], [(2,3),(1,0)], [(3,1),(0,2)], [(2,2),(1,1)], [(1,3),(2,0)], [(2,1),(1,2)], [(2,1),(1,1),(0,1)], [(3,0),(0,2),(0,1)], [(2,2),(1,0),(0,1)], [(2,1),(0,2),(1,0)], [(1,2),(2,0),(0,1)], [(1,2),(1,1),(1,0)], [(0,3),(2,0),(1,0)], [(2,0),(1,1),(0,2)], [(2,0),(0,2),(1,0),(0,1)].
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 2, 5, 15, 52, 203, ...
%e 1, 1, 5, 40, 457, 6995, 136771, ...
%e 1, 2, 17, 364, 14595, 937776, 88507276, ...
%e 1, 2, 46, 2897, 407287, 107652681, ...
%e 1, 3, 123, 21369, 10200931, 10781201973, ...
%e 1, 4, 323, 148257, 233051939, ...
%e 1, 5, 809, 970246, 4909342744, ...
%t f[n_, k_] := f[n, k] = 1/2 Product[Sum[O[x[j]]^(n+1), {j, 1, k}]+1+ Product[x[j]^i[j], {j, 1, k}], Evaluate[Sequence @@ Table[{i[j], 0, n}, {j, 1, k}]]];
%t a[0, _] = a[_, 0] = 1; a[n_, k_] := SeriesCoefficient[f[n, k], Sequence @@ Table[{x[j], 0, n}, {j, 1, k}]];
%t Table[Print[a[n-k, k]]; a[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 11 2013, updated Sep 16 2019 *)
%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+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)} \\ _Andrew Howroyd_, Dec 16 2018
%Y Columns k=0..5 give: A000012, A000009, A219554, A219560, A219561, A219565.
%Y Rows n=0..3 give: A000012, A000110, A094574, A319591.
%Y Cf. A188445, A219727 (partitions of {n}^k into k-tuples), A318286.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Nov 23 2012
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