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%I #18 Sep 24 2019 08:17:11
%S 1,52,6995,937776,107652681,10781201973,958919976957,76861542428397,
%T 5620227129073491,378709513816248475,23713852762539359688,
%U 1389561695379881634055,76647024053735036288641,3999799865715906390697377,198328846122797866982616805,9379277765981012067789260214
%N Number of 5-partite partitions of (n,n,n,n,n) into distinct quintuples.
%C Number of factorizations of (p*q*r*s*t)^n into distinct factors where p, q, r, s, t are distinct primes.
%F a(n) = [(v*w*x*y*z)^n] 1/2 * Product_{h,i,j,k,m>=0} (1+v^h*w^i*x^j*y^k*z^m).
%t a[n_] := a[n] = If[n == 0, 1, (1/2) Coefficient[Product[O[v]^(n+1) + O[w]^(n+1) + O[x]^(n+1) + O[y]^(n+1) + O[z]^(n+1) + (1 + v^i w^j x^k y^l z^m), {i, 0, n}, {j, 0, n}, {k, 0, n}, {l, 0, n}, {m, 0, n}] // Normal, (v w x y z)^n]];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* _Jean-François Alcover_, Sep 24 2019 *)
%Y Column k=5 of A219585.
%Y Cf. A002774, A219554, A219560, A219561.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Nov 23 2012
%E a(6) from _Alois P. Heinz_, Sep 25 2014
%E a(7)-a(15) from _Andrew Howroyd_, Dec 16 2018
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