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A256384 Number A(n,k) of factorizations of m^n into at most n factors, where m is a product of exactly k distinct primes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A256384 #19 Oct 26 2018 16:15:09
%S A256384 1,1,1,1,1,1,1,1,2,1,1,1,5,3,1,1,1,14,19,5,1,1,1,41,171,74,7,1,1,1,
%T A256384 122,1675,1975,248,11,1,1,1,365,16683,64182,20096,814,15,1,1,1,1094,
%U A256384 166699,2203215,2213016,187921,2457,22,1,1,1,3281,1666731,76727374,268446852,69406700,1609727,7168,30,1
%N A256384 Number A(n,k) of factorizations of m^n into at most n factors, where m is a product of exactly k distinct primes; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A256384 A(n,k) is also the number of k-partite partitions of (n)^k into at most n k-tuples.  A(2,2) = 5: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(1,2),(1,0)], [(1,1),(1,1)].
%e A256384 A(2,2) = 5: (2*3)^2 = 36 has 5 factorizations into at most 2 factors: 36, 2*18, 3*12, 4*9, 6*6.
%e A256384 Square array A(n,k) begins:
%e A256384   1, 1,   1,     1,       1,         1, ...
%e A256384   1, 1,   1,     1,       1,         1, ...
%e A256384   1, 2,   5,    14,      41,       122, ...
%e A256384   1, 3,  19,   171,    1675,     16683, ...
%e A256384   1, 5,  74,  1975,   64182,   2203215, ...
%e A256384   1, 7, 248, 20096, 2213016, 268446852, ...
%t A256384 b[n_, k_, i_] := b[n, k, i] = If[n>k, 0, 1] + If[PrimeQ[n] || i<2, 0, Sum[ If[d > k, 0, b[n/d, d, i-1]], {d, Divisors[n][[2 ;; -2]]}]]; A[0, _] = 1; A[1, _] = 1; A[_, 0] = 1; A[n_, k_] := With[{t = Times @@ Prime[ Range[k] ]}, b[t^n, t^n, n]]; Table[diag = Table[A[n-k, k], {k, n, 0, -1}]; Print[ diag]; diag, {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 08 2016, after _Alois P. Heinz_ *)
%Y A256384 Columns k=0-3 give: A000012, A000041, A254686, A254811.
%Y A256384 Rows n=0+1,2-3 give: A000012, A007051, A256493.
%Y A256384 Cf. A219727.
%K A256384 nonn,tabl
%O A256384 0,9
%A A256384 _Alois P. Heinz_, Mar 27 2015