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A219561 Number of 4-partite partitions of (n,n,n,n) into distinct quadruples. 5
1, 15, 457, 14595, 407287, 10200931, 233051939, 4909342744, 96272310302, 1771597038279, 30795582025352, 508466832109216, 8011287089600483, 120926718707154007, 1754672912487450236, 24547188914867491083, 331937179344717327559, 4348524173437743243649, 55300773426746984710983 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Sep 16 2019 *)

CROSSREFS

Column k=4 of A219585.

Cf. A002774, A219554, A219560, A219565.

Sequence in context: A041420 A036506 A306675 * A209488 A320097 A177080

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

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Last modified May 21 06:36 EDT 2022. Contains 353889 sequences. (Running on oeis4.)