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

6,2

Links

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

Wikipedia, Partition (number theory)

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

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 *)

Crossrefs

Column k=6 of A308680.

Sequence in context: A100189 A052267 A038166 * A121596 A264026 A341385

Adjacent sequences: A327381 A327382 A327383 * A327385 A327386 A327387

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved