A327385 Number of colored integer partitions of n such that seven colors are used and parts differ by size or by color.

1, 7, 35, 133, 434, 1253, 3311, 8135, 18851, 41573, 87920, 179305, 354270, 680631, 1275430, 2337097, 4196717, 7398699, 12826324, 21895160, 36848119, 61201709, 100415175, 162886318, 261422357, 415397836, 653899589, 1020282424, 1578729491, 2423647471, 3693050242

Offset: 7

Author

Alois P. Heinz, Sep 03 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 7..10000 (terms n = 5001..9000 from Vaclav Kotesovec)
• Wikipedia, Partition (number theory)

Crossrefs

Column k=7 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))(7):
seq(a(n), n=7..45);

Mathematica

A327385[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^7, {x, 0, n}]; Table[A327385[n], {n, 7, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

Formula

a(n) ~ exp(Pi*sqrt(7*n/3)) * 7^(1/4) / (32 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^7. - Ilya Gutkovskiy, Jan 31 2021