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.

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, 25, 3, 1, 1, 1, 1, 71, 566, 465, 49, 4, 1, 1, 1, 1, 197, 4781, 11131, 2505, 103, 4, 1, 1, 1, 1, 554, 41357, 297381, 190131, 12652, 184, 5, 1, 1, 1, 1, 1570, 364470, 8349223, 16669641, 2928876, 57232, 331, 5, 1, 1

Offset: 0

Author

Alois P. Heinz, Apr 24 2015

Keywords

• nonn
• tabl

Comments

Also number of ways to partition the multiset consisting of n copies each of 1, 2, ..., k into n multisets of size k.

Examples

			A(4,2) = 3: (2*3)^4 = 1296 = 6*6*6*6 = 9*6*6*4 = 9*9*4*4.
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.
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.
Square array A(n,k) begins:
  1, 1, 1,  1,    1,      1, ...
  1, 1, 1,  1,    1,      1, ...
  1, 1, 2,  4,   10,     26, ...
  1, 1, 2, 10,   70,    566, ...
  1, 1, 3, 25,  465,  11131, ...
  1, 1, 3, 49, 2505, 190131, ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals n=0..26)

Crossrefs

Columns k=0+1, 2-5 give: A000012, A008619, A254233, A257114, A257518.

Rows n=0+1, 2-3 give: A000012, A257520, A333902.

Main diagonal gives A334286.

Cf. A257463, A333901 (ordered factorizations).

Programs

Maple

with(numtheory):
b:= proc(n, i, k) option remember; `if`(n=1, 1,
      add(`if`(d>i or bigomega(d)<>k, 0,
      b(n/d, d, k)), d=divisors(n) minus {1}))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..k)^n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

Mathematica

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 *)