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

7,2

Links

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

Wikipedia, Partition (number theory)

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

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 *)

Crossrefs

Column k=7 of A308680.

Sequence in context: A328019 A327243 A059595 * A344101 A001941 A320050

Adjacent sequences: A327382 A327383 A327384 * A327386 A327387 A327388

Keyword

nonn

Author

Alois P. Heinz, Sep 03 2019

Status

approved