A257463 Number A(n,k) of factorizations of m^k into n factors, where m is a product of exactly n 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, 2, 5, 1, 1, 1, 1, 3, 10, 17, 1, 1, 1, 1, 3, 23, 93, 73, 1, 1, 1, 1, 4, 40, 465, 1417, 388, 1, 1, 1, 1, 4, 73, 1746, 19834, 32152, 2461, 1, 1, 1, 1, 5, 114, 5741, 190131, 1532489, 1016489, 18155, 1, 1

Offset

0,13

Comments

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

Links

Andrew Howroyd, Antidiagonals n = 0..27, flattened (antidiagonals 0..12 from Alois P. Heinz)

P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. The array is on page 40.

Math StackExchange, Number of ways to partition 40 balls with 4 colors into 4 baskets

Marko Riedel, Maple program to compute array from cycle indices

Example

A(4,2) = 17: (2*3*5*7)^2 = 44100 = 15*15*14*14 = 21*15*14*10 = 21*21*10*10 = 25*14*14*9 = 25*21*14*6 = 25*21*21*4 = 35*14*10*9 = 35*15*14*6 = 35*21*10*6 = 35*21*15*4 = 35*35*6*6 = 35*35*9*4 = 49*10*10*9 = 49*15*10*6 = 49*15*15*4 = 49*25*6*6 = 49*25*9*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) = 3: (2*3)^4 = 1296 = 36*36 = 54*24 = 81*16.
Square array A(n,k) begins:
  1, 1,   1,     1,       1,        1,         1, ...
  1, 1,   1,     1,       1,        1,         1, ...
  1, 1,   2,     2,       3,        3,         4, ...
  1, 1,   5,    10,      23,       40,        73, ...
  1, 1,  17,    93,     465,     1746,      5741, ...
  1, 1,  73,  1417,   19834,   190131,   1398547, ...
  1, 1, 388, 32152, 1532489, 43816115, 848597563, ...

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)))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k$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, DivisorSum[n, If[#>i || PrimeOmega[#] != k, 0, b[n/#, #, k]]&]];
A[n_, k_] := b[p = Product[Prime[i], {i, 1, n}]^k, p, k];
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *)

Crossrefs

Columns k=0+1, 2-4 give: A000012, A002135, A254243, A268668.

Rows n=0+1, 2-5 give: A000012, A008619, A257464, A253259, A253263.

Main diagonal gives A334286.

Cf. A257462, A257493 (ordered factorizations).

Sequence in context: A265313 A106498 A093466 * A293483 A125761 A154950

Adjacent sequences: A257460 A257461 A257462 * A257464 A257465 A257466

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 24 2015

Status

approved