A327387 Number of colored integer partitions of n such that nine colors are used and parts differ by size or by color.

1, 9, 54, 246, 945, 3186, 9729, 27414, 72315, 180415, 429156, 979425, 2155485, 4593330, 9510624, 19188360, 37815948, 72950634, 138002024, 256405887, 468550278, 843138585, 1495634373, 2617905474, 4525424256, 7731765279, 13065217956, 21849902348, 36184992984

Offset: 9

Author

Alois P. Heinz, Sep 03 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..10000
• Wikipedia, Partition (number theory)

Crossrefs

Column k=9 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))(9):
seq(a(n), n=9..45);

Mathematica

A327387[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^9, {x, 0, n}]; Table[A327387[n], {n, 9, 37}] (* Robert P. P. McKone, Jan 31 2021 *)

Formula

a(n) ~ exp(Pi*sqrt(3*n)) * 3^(1/4) / (64 * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

G.f.: (-1 + Product_{k>=1} (1 + x^k))^9. - Ilya Gutkovskiy, Jan 31 2021