A219561 Number of 4-partite partitions of (n,n,n,n) into distinct quadruples.

1, 15, 457, 14595, 407287, 10200931, 233051939, 4909342744, 96272310302, 1771597038279, 30795582025352, 508466832109216, 8011287089600483, 120926718707154007, 1754672912487450236, 24547188914867491083, 331937179344717327559, 4348524173437743243649, 55300773426746984710983

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..18.

Formula

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

Example

a(0) = 1: [].
a(1) = 15: [(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)], [(0,0,1,1),(1,0,0,0),(0,1,0,0)], [(0,1,0,1),(1,0,0,0),(0,0,1,0)], [(0,1,1,0),(1,0,0,0),(0,0,0,1)], [(1,0,0,1),(0,1,0,0),(0,0,1,0)], [(1,0,0,1),(0,1,1,0)], [(1,0,1,0),(0,1,0,0),(0,0,0,1)], [(1,0,1,0),(0,1,0,1)], [(1,1,0,0),(0,0,1,0),(0,0,0,1)], [(1,1,0,0),(0,0,1,1)], [(0,1,1,1),(1,0,0,0)], [(1,0,1,1),(0,1,0,0)], [(1,1,0,1),(0,0,1,0)], [(1,1,1,0),(0,0,0,1)], [(1,1,1,1)].

Mathematica

a[n_] := If[n == 0, 1, (1/2) Coefficient[Product[O[w]^(n+1) + O[x]^(n+1) + O[y]^(n+1) + O[z]^(n+1) + (1 + w^i x^j y^k z^m), {i, 0, n}, {j, 0, n}, {k, 0, n}, {m, 0, n}] // Normal, (w x y z)^n]];
Table[Print[n]; a[n], {n, 0, 12}] (* Jean-François Alcover, Sep 16 2019 *)

Crossrefs

Column k=4 of A219585.

Cf. A002774, A219554, A219560, A219565.

Sequence in context: A036506 A369828 A306675 * A209488 A361307 A320097

Adjacent sequences: A219558 A219559 A219560 * A219562 A219563 A219564

Keyword

nonn

Author

Alois P. Heinz, Nov 23 2012

Extensions

a(9) from Alois P. Heinz, Oct 15 2014

a(10)-a(18) from Andrew Howroyd, Dec 17 2018

Status

approved