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A327388 Number of colored integer partitions of n such that ten colors are used and parts differ by size or by color.

%I A327388 #19 Jan 31 2021 16:51:25
%S A327388 1,10,65,320,1320,4762,15500,46410,129710,341990,857695,2059430,
%T A327388 4759235,10630810,23034880,48562378,99866045,200766810,395317950,
%U A327388 763661010,1449390299,2706189810,4976391015,9021860260,16139848000,28515535112,49792637480,85989053350
%N A327388 Number of colored integer partitions of n such that ten colors are used and parts differ by size or by color.
%C A327388 In general, column k > 0 of A308680 is asymptotic to exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - _Vaclav Kotesovec_, Sep 16 2019
%H A327388 Alois P. Heinz, <a href="/A327388/b327388.txt">Table of n, a(n) for n = 10..10000</a> (terms n = 5001..8000 from Vaclav Kotesovec)
%H A327388 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327388 a(n) ~ exp(Pi*sqrt(10*n/3)) * 5^(1/4) / (2^(25/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 16 2019
%F A327388 G.f.: (-1 + Product_{k>=1} (1 + x^k))^10. - _Ilya Gutkovskiy_, Jan 31 2021
%p A327388 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A327388       b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
%p A327388     end:
%p A327388 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(10):
%p A327388 seq(a(n), n=10..45);
%t A327388 A327388[n_] := SeriesCoefficient[(Product[(1 + x^k), {k, 1, n}] - 1)^10, {x, 0, n}]; Table[A327388[n], {n, 10, 37}] (* _Robert P. P. McKone_, Jan 31 2021 *)
%Y A327388 Column k=10 of A308680.
%K A327388 nonn
%O A327388 10,2
%A A327388 _Alois P. Heinz_, Sep 03 2019