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A327384 Number of colored integer partitions of n such that six colors are used and parts differ by size or by color.

%I A327384 #16 Jan 31 2021 12:33:18
%S A327384 1,6,27,92,273,720,1751,3978,8565,17618,34878,66792,124268,225384,
%T A327384 399618,694294,1184340,1986900,3282991,5349372,8604978,13678190,
%U A327384 21503439,33459222,51563824,78751470,119259576,179169140,267154842,395521482,581629358,849846186
%N A327384 Number of colored integer partitions of n such that six colors are used and parts differ by size or by color.
%H A327384 Vaclav Kotesovec, <a href="/A327384/b327384.txt">Table of n, a(n) for n = 6..10000</a> (terms 6..5000 from Alois P. Heinz)
%H A327384 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327384 a(n) ~ exp(Pi*sqrt(2*n)) / (2^(17/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 16 2019
%F A327384 G.f.: (-1 + Product_{k>=1} (1 + x^k))^6. - _Ilya Gutkovskiy_, Jan 31 2021
%p A327384 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A327384       b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
%p A327384     end:
%p A327384 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(6):
%p A327384 seq(a(n), n=6..45);
%t A327384 A327384[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^6, {x, 0, n}]; Table[A327384[n], {n, 6, 37}] (* _Robert P. P. McKone_, Jan 31 2021 *)
%Y A327384 Column k=6 of A308680.
%K A327384 nonn
%O A327384 6,2
%A A327384 _Alois P. Heinz_, Sep 03 2019