A219565 Number of 5-partite partitions of (n,n,n,n,n) into distinct quintuples.

1, 52, 6995, 937776, 107652681, 10781201973, 958919976957, 76861542428397, 5620227129073491, 378709513816248475, 23713852762539359688, 1389561695379881634055, 76647024053735036288641, 3999799865715906390697377, 198328846122797866982616805, 9379277765981012067789260214

Offset: 0

Author

Alois P. Heinz, Nov 23 2012

Keywords

• nonn

Comments

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

Crossrefs

Column k=5 of A219585.

Cf. A002774, A219554, A219560, A219561.

Programs

Mathematica

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]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 7}] (* Jean-François Alcover, Sep 24 2019 *)

Formula

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).

Extensions

a(6) from Alois P. Heinz, Sep 25 2014

a(7)-a(15) from Andrew Howroyd, Dec 16 2018