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
Keywords
Examples
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)].
Programs
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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 *)
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).
Extensions
a(9) from Alois P. Heinz, Oct 15 2014
a(10)-a(18) from Andrew Howroyd, Dec 17 2018
Comments