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%I #21 Jan 15 2016 04:02:19
%S 1,5,40,364,2897,21369,148257,970246,6032341,35850410,204646488,
%T 1126463948,5999145787,30999381232,155798366059,763194776551,
%U 3650648583934,17079277343463,78262895082681,351708874155894,1551843168854346
%N Number of tripartite partitions of (n,n,n) into distinct triples.
%C Number of factorizations of (p*q*r)^n into distinct factors where p, q, r are distinct primes.
%F a(n) = [(x*y*z)^n] 1/2 * Product_{i,j,k>=0} (1+x^i*y^j*z^k).
%e a(0) = 1: [].
%e 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)].
%p with(numtheory):
%p b:= proc(n, k) option remember;
%p `if`(n>k, 0, 1) +`if`(isprime(n), 0,
%p add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
%p end:
%p a:= n-> b(30^n$2):
%p seq(a(n), n=0..10); # _Alois P. Heinz_, May 26 2013
%t 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_ *)
%Y Column k=3 of A219585.
%Y Cf. A002774, A219554, A219561, A219565, A219678.
%K nonn,more
%O 0,2
%A _Alois P. Heinz_, Nov 23 2012
%E a(16) from _Alois P. Heinz_, May 26 2013
%E a(17) from _Alois P. Heinz_, Sep 24 2014
%E More terms from _Jean-François Alcover_, Jan 15 2016
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