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

Author

Alois P. Heinz, Sep 03 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 8..10000
• Wikipedia, Partition (number theory)

Crossrefs

Column k=8 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))(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 *)

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