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A327385
Number of colored integer partitions of n such that seven colors are used and parts differ by size or by color.
6
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)
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 03 2019
STATUS
approved