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

0,2

Comments

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

Links

Table of n, a(n) for n=0..20.

Formula

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

Example

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)].

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 *)

Crossrefs

Column k=3 of A219585.

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

Sequence in context: A371520 A052788 A213104 * A380210 A349362 A359984

Adjacent sequences: A219557 A219558 A219559 * A219561 A219562 A219563

Keyword

nonn,more

Author

Alois P. Heinz, Nov 23 2012

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

Status

approved