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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.

%I A327385 #19 Jan 31 2021 12:33:25
%S A327385 1,7,35,133,434,1253,3311,8135,18851,41573,87920,179305,354270,680631,
%T A327385 1275430,2337097,4196717,7398699,12826324,21895160,36848119,61201709,
%U A327385 100415175,162886318,261422357,415397836,653899589,1020282424,1578729491,2423647471,3693050242
%N A327385 Number of colored integer partitions of n such that seven colors are used and parts differ by size or by color.
%H A327385 Alois P. Heinz, <a href="/A327385/b327385.txt">Table of n, a(n) for n = 7..10000</a> (terms n = 5001..9000 from Vaclav Kotesovec)
%H A327385 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327385 a(n) ~ exp(Pi*sqrt(7*n/3)) * 7^(1/4) / (32 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 16 2019
%F A327385 G.f.: (-1 + Product_{k>=1} (1 + x^k))^7. - _Ilya Gutkovskiy_, Jan 31 2021
%p A327385 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A327385       b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
%p A327385     end:
%p A327385 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(7):
%p A327385 seq(a(n), n=7..45);
%t A327385 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 *)
%Y A327385 Column k=7 of A308680.
%K A327385 nonn
%O A327385 7,2
%A A327385 _Alois P. Heinz_, Sep 03 2019