A219560 Number of tripartite partitions of (n,n,n) into distinct triples.

1, 5, 40, 364, 2897, 21369, 148257, 970246, 6032341, 35850410, 204646488, 1126463948, 5999145787, 30999381232, 155798366059, 763194776551, 3650648583934, 17079277343463, 78262895082681, 351708874155894, 1551843168854346

Offset: 0

Author

Alois P. Heinz, Nov 23 2012

Keywords

• nonn
• more

Comments

Number of factorizations of (p*q*r)^n into distinct factors where p, q, r are distinct primes.

Examples

			a(0) = 1: [].
a(1) = 5: [(1,1,1)], [(1,1,0),(0,0,1)], [(1,0,1),(0,1,0)], [(0,1,1),(1,0,0)], [(0,0,1),(0,1,0),(1,0,0)].
		

Crossrefs

Column k=3 of A219585.

Cf. A002774, A219554, A219561, A219565, A219678.

Programs

Maple

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(30^n$2):
seq(a(n), n=0..10);  # Alois P. Heinz, May 26 2013

Mathematica

b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, b[n/d, d - 1]], {d, Divisors[n][[2 ;; -2]]}]]; a[0] = 1; a[n_] := b[30^n, 30^n];  Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

Formula

a(n) = [(x*y*z)^n] 1/2 * Product_{i,j,k>=0} (1+x^i*y^j*z^k).

Extensions

a(16) from Alois P. Heinz, May 26 2013

a(17) from Alois P. Heinz, Sep 24 2014

More terms from Jean-François Alcover, Jan 15 2016