A327386 Number of colored integer partitions of n such that eight colors are used and parts differ by size or by color.

1, 8, 44, 184, 654, 2048, 5836, 15400, 38173, 89752, 201740, 436104, 911072, 1846648, 3643360, 7016016, 13217634, 24408992, 44260816, 78923480, 138571450, 239838288, 409619196, 690956800, 1152075322, 1900139104, 3102050748, 5015671600, 8036376650, 12766039888

Offset

8,2

Links

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Wikipedia, Partition (number theory)

Formula

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(19/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^8. - 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))(8):
seq(a(n), n=8..45);

Mathematica

A327386[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^8, {x, 0, n}]; Table[A327386[n], {n, 8, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

Crossrefs

Column k=8 of A308680.

Sequence in context: A165618 A250285 A059596 * A341386 A181358 A005798

Adjacent sequences: A327383 A327384 A327385 * A327387 A327388 A327389

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved