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A327381 Number of colored integer partitions of n such that three colors are used and parts differ by size or by color.

%I A327381 #18 Jan 29 2021 19:02:31
%S A327381 1,3,9,19,39,72,128,216,354,563,876,1335,1998,2946,4284,6154,8742,
%T A327381 12294,17129,23667,32442,44151,59682,80169,107054,142167,187812,
%U A327381 246895,323058,420852,545958,705438,908043,1164609,1488504,1896174,2407836,3048255,3847716
%N A327381 Number of colored integer partitions of n such that three colors are used and parts differ by size or by color.
%C A327381 With offset 0 convolution cube of A000009(k+1). - _George Beck_, Jan 29 2021
%H A327381 Vaclav Kotesovec, <a href="/A327381/b327381.txt">Table of n, a(n) for n = 3..10000</a> (terms 3..5000 from Alois P. Heinz)
%H A327381 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327381 a(n) ~ exp(Pi*sqrt(n)) / (8 * n^(3/4)). - _Vaclav Kotesovec_, Sep 14 2019
%F A327381 G.f.: (-1 + Product_{m >= 1} (1 + x^m))^3. - _George Beck_, Jan 29 2021
%p A327381 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A327381       b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
%p A327381     end:
%p A327381 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
%p A327381 seq(a(n), n=3..45);
%t A327381 b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k] Binomial[k, j]], {j, 0, Min[k, n/i]}]]];
%t A327381 a[n_] := With[{k = 3}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t A327381 a /@ Range[3, 45] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y A327381 Column k=3 of A308680.
%Y A327381 Cf. A000009.
%K A327381 nonn
%O A327381 3,2
%A A327381 _Alois P. Heinz_, Sep 03 2019