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A327387 Number of colored integer partitions of n such that nine colors are used and parts differ by size or by color.

%I A327387 #17 Jan 31 2021 12:33:39
%S A327387 1,9,54,246,945,3186,9729,27414,72315,180415,429156,979425,2155485,
%T A327387 4593330,9510624,19188360,37815948,72950634,138002024,256405887,
%U A327387 468550278,843138585,1495634373,2617905474,4525424256,7731765279,13065217956,21849902348,36184992984
%N A327387 Number of colored integer partitions of n such that nine colors are used and parts differ by size or by color.
%H A327387 Alois P. Heinz, <a href="/A327387/b327387.txt">Table of n, a(n) for n = 9..10000</a>
%H A327387 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327387 a(n) ~ exp(Pi*sqrt(3*n)) * 3^(1/4) / (64 * n^(3/4)). - _Vaclav Kotesovec_, Sep 16 2019
%F A327387 G.f.: (-1 + Product_{k>=1} (1 + x^k))^9. - _Ilya Gutkovskiy_, Jan 31 2021
%p A327387 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A327387       b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
%p A327387     end:
%p A327387 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(9):
%p A327387 seq(a(n), n=9..45);
%t A327387 A327387[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^9, {x, 0, n}]; Table[A327387[n], {n, 9, 37}] (* _Robert P. P. McKone_, Jan 31 2021 *)
%Y A327387 Column k=9 of A308680.
%K A327387 nonn
%O A327387 9,2
%A A327387 _Alois P. Heinz_, Sep 03 2019