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A327386 Number of colored integer partitions of n such that eight colors are used and parts differ by size or by color.

%I A327386 #16 Jan 31 2021 12:33:32
%S A327386 1,8,44,184,654,2048,5836,15400,38173,89752,201740,436104,911072,
%T A327386 1846648,3643360,7016016,13217634,24408992,44260816,78923480,
%U A327386 138571450,239838288,409619196,690956800,1152075322,1900139104,3102050748,5015671600,8036376650,12766039888
%N A327386 Number of colored integer partitions of n such that eight colors are used and parts differ by size or by color.
%H A327386 Alois P. Heinz, <a href="/A327386/b327386.txt">Table of n, a(n) for n = 8..10000</a>
%H A327386 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327386 a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(19/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 16 2019
%F A327386 G.f.: (-1 + Product_{k>=1} (1 + x^k))^8. - _Ilya Gutkovskiy_, Jan 31 2021
%p A327386 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A327386       b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
%p A327386     end:
%p A327386 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(8):
%p A327386 seq(a(n), n=8..45);
%t A327386 A327386[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^8, {x, 0, n}]; Table[A327386[n], {n, 8, 37}] (* _Robert P. P. McKone_, Jan 31 2021 *)
%Y A327386 Column k=8 of A308680.
%K A327386 nonn
%O A327386 8,2
%A A327386 _Alois P. Heinz_, Sep 03 2019