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A385520 Expansion of Product_{k>0} ((1 - q^(2*k))*(1 - q^(6*k))^3)/((1 - q^k)*(1 - q^(3*k))*(1 - q^(4*k))*(1 - q^(12*k))).

%I A385520 #23 Jul 19 2025 12:59:46
%S A385520 1,1,1,3,4,5,6,9,13,16,20,27,36,44,54,69,88,107,130,162,200,240,288,
%T A385520 351,426,507,602,723,864,1019,1200,1422,1681,1968,2300,2700,3160,3674,
%U A385520 4266,4965,5768,6665,7692,8892,10260,11792,13536,15552,17844,20407
%N A385520 Expansion of Product_{k>0} ((1 - q^(2*k))*(1 - q^(6*k))^3)/((1 - q^k)*(1 - q^(3*k))*(1 - q^(4*k))*(1 - q^(12*k))).
%C A385520 a(n) is the number of partitions of the integer n wherein each even part can appear at most twice, while each odd part can appear once, thrice, or four times.
%C A385520 Also, a(n) is the unsigned version of the sequence given in A293306 (this can be seen by replacing q by -q in the generating function).
%H A385520 Seiichi Manyama, <a href="/A385520/b385520.txt">Table of n, a(n) for n = 0..10000</a>
%H A385520 Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, <a href="https://arxiv.org/abs/2507.10965">Convolutive sequences, I: Through the lens of integer partition functions</a>, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 15.
%F A385520 G.f.: Product_{k>0} ((1 - q^(2*k))*(1 - q^(6*k))^3)/((1 - q^k)*(1 - q^(3*k))*(1 - q^(4*k))*(1 - q^(12*k))).
%F A385520 G.f.: (eta(-q)*eta(-q^3))/eta(q^2)^2.
%F A385520 a(n) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)). - _Vaclav Kotesovec_, Jul 02 2025
%e A385520 For n = 4, the a(4) = 4 partitions are 4, 3+1, 2+2, and 1+1+1+1.  Note that there is one other partition of 4 which is NOT counted by a(4); that is the partition 2+1+1. This partition is NOT counted by a(4) because the odd part 1 appears twice, and this is not allowed from the description given above.
%p A385520 p:=product((1-q^(2*k))*(1-q^(6*k))^3/((1-q^k)*(1-q^(3*k))*(1-q^(4*k))*(1-q^(12*k))), k=1..1000): s:=series(p,q,1000):
%t A385520 nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k)) * (1 - x^(6*k))^3 / ((1 - x^k) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(12*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 02 2025 *)
%o A385520 (PARI) my(N=50,q='q+O('q^N)); Vec((eta(-q)*eta(-q^3))/eta(q^2)^2) \\ _Joerg Arndt_, Jul 02 2025
%Y A385520 Cf. A293306.
%K A385520 nonn
%O A385520 0,4
%A A385520 _James Sellers_, Jul 01 2025