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

9,2

Links

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Wikipedia, Partition (number theory)

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

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 *)

Crossrefs

Column k=9 of A308680.

Sequence in context: A250286 A289254 A059597 * A282920 A023008 A079817

Adjacent sequences: A327384 A327385 A327386 * A327388 A327389 A327390

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved