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A327382 Number of colored integer partitions of n such that four colors are used and parts differ by size or by color.

%I A327382 #18 Jan 29 2021 19:41:02
%S A327382 1,4,14,36,85,180,360,680,1234,2160,3674,6092,9882,15724,24594,37884,
%T A327382 57553,86344,128060,187948,273178,393516,562158,796860,1121375,
%U A327382 1567336,2176664,3004692,4124130,5630160,7646916,10335696,13905376,18625564,24843142,33003072
%N A327382 Number of colored integer partitions of n such that four colors are used and parts differ by size or by color.
%C A327382 With offset 0 four-fold convolution of A000009(k+1). - _George Beck_, Jan 29 2021
%H A327382 Vaclav Kotesovec, <a href="/A327382/b327382.txt">Table of n, a(n) for n = 4..10000</a> (terms 4..5000 from Alois P. Heinz)
%H A327382 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327382 a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 14 2019
%F A327382 G.f.: (-1 + Product_{m >= 1} (1 + x^m))^4. - _George Beck_, Jan 29 2021
%p A327382 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A327382       b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
%p A327382     end:
%p A327382 a:= n-> (k-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k))(4):
%p A327382 seq(a(n), n=4..45);
%t A327382 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 A327382 a[n_] := With[{k = 4}, Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}]];
%t A327382 a /@ Range[4, 45] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y A327382 Column k=4 of A308680.
%Y A327382 Cf. A000009.
%K A327382 nonn
%O A327382 4,2
%A A327382 _Alois P. Heinz_, Sep 03 2019