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A257462 Number A(n,k) of factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A257462 #32 Apr 21 2020 18:25:27
%S A257462 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,4,2,1,1,1,1,10,10,3,1,1,1,1,26,70,
%T A257462 25,3,1,1,1,1,71,566,465,49,4,1,1,1,1,197,4781,11131,2505,103,4,1,1,1,
%U A257462 1,554,41357,297381,190131,12652,184,5,1,1,1,1,1570,364470,8349223,16669641,2928876,57232,331,5,1,1
%N A257462 Number A(n,k) of factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A257462 Also number of ways to partition the multiset consisting of n copies each of 1, 2, ..., k into n multisets of size k.
%H A257462 Andrew Howroyd, <a href="/A257462/b257462.txt">Table of n, a(n) for n = 0..377</a> (antidiagonals n=0..26)
%e A257462 A(4,2) = 3: (2*3)^4 = 1296 = 6*6*6*6 = 9*6*6*4 = 9*9*4*4.
%e A257462 A(3,3) = 10: (2*3*5)^3 = 2700 = 30*30*30 = 45*30*20 = 50*27*20 = 50*30*18 = 50*45*12 = 75*20*18 = 75*30*12 = 75*45*8 = 125*18*12 = 125*27*8.
%e A257462 A(2,4) = 10: (2*3*5*7)^2 = 44100 = 210*210 = 225*196 = 294*150 = 315*140 = 350*126 = 441*100 = 490*90 = 525*84 = 735*60 = 1225*36.
%e A257462 Square array A(n,k) begins:
%e A257462   1, 1, 1,  1,    1,      1, ...
%e A257462   1, 1, 1,  1,    1,      1, ...
%e A257462   1, 1, 2,  4,   10,     26, ...
%e A257462   1, 1, 2, 10,   70,    566, ...
%e A257462   1, 1, 3, 25,  465,  11131, ...
%e A257462   1, 1, 3, 49, 2505, 190131, ...
%p A257462 with(numtheory):
%p A257462 b:= proc(n, i, k) option remember; `if`(n=1, 1,
%p A257462       add(`if`(d>i or bigomega(d)<>k, 0,
%p A257462       b(n/d, d, k)), d=divisors(n) minus {1}))
%p A257462     end:
%p A257462 A:= (n, k)-> b(mul(ithprime(i), i=1..k)^n$2, k):
%p A257462 seq(seq(A(n, d-n), n=0..d), d=0..8);
%t A257462 b[n_, i_, k_] := b[n, i, k] = If[n == 1, 1, Sum[If[d > i || PrimeOmega[d] != k, 0, b[n/d, d, k]], {d, Divisors[n] // Rest}]]; A[n_, k_] := Module[ {p = Product[Prime[i], {i, 1, k}]^n}, b[p, p, k]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 8}] // Flatten (* _Jean-François Alcover_, Feb 22 2016, after _Alois P. Heinz_ *)
%Y A257462 Columns k=0+1, 2-5 give: A000012, A008619, A254233, A257114, A257518.
%Y A257462 Rows n=0+1, 2-3 give: A000012, A257520, A333902.
%Y A257462 Main diagonal gives A334286.
%Y A257462 Cf. A257463, A333901 (ordered factorizations).
%K A257462 nonn,tabl
%O A257462 0,13
%A A257462 _Alois P. Heinz_, Apr 24 2015