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A219560 Number of tripartite partitions of (n,n,n) into distinct triples. 6
1, 5, 40, 364, 2897, 21369, 148257, 970246, 6032341, 35850410, 204646488, 1126463948, 5999145787, 30999381232, 155798366059, 763194776551, 3650648583934, 17079277343463, 78262895082681, 351708874155894, 1551843168854346 (list; graph; refs; listen; history; text; internal format)
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: A137973 A052788 A213104 * A349362 A271957 A220673

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

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Last modified January 22 16:25 EST 2022. Contains 350483 sequences. (Running on oeis4.)