A327384 Number of colored integer partitions of n such that six colors are used and parts differ by size or by color.

1, 6, 27, 92, 273, 720, 1751, 3978, 8565, 17618, 34878, 66792, 124268, 225384, 399618, 694294, 1184340, 1986900, 3282991, 5349372, 8604978, 13678190, 21503439, 33459222, 51563824, 78751470, 119259576, 179169140, 267154842, 395521482, 581629358, 849846186

Offset: 6

Author

Alois P. Heinz, Sep 03 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 6..10000 (terms 6..5000 from Alois P. Heinz)
• Wikipedia, Partition (number theory)

Crossrefs

Column k=6 of A308680.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(6):
seq(a(n), n=6..45);

Mathematica

A327384[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^6, {x, 0, n}]; Table[A327384[n], {n, 6, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

Formula

a(n) ~ exp(Pi*sqrt(2*n)) / (2^(17/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^6. - Ilya Gutkovskiy, Jan 31 2021