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A302834 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^3)).

%I A302834 #10 Apr 14 2018 14:39:49
%S A302834 1,2,3,4,5,6,7,8,10,12,14,16,18,20,22,24,27,30,33,36,39,42,45,48,52,
%T A302834 56,60,65,70,75,80,85,91,97,103,110,117,124,131,138,146,154,162,171,
%U A302834 180,189,198,207,217,227,237,248,259,270,282,294,307,320,333,347,361,375,390,405,422
%N A302834 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^3)).
%C A302834 Partial sums of A003108.
%C A302834 Number of partitions of n into cubes if there are two kinds of 1's.
%H A302834 Alois P. Heinz, <a href="/A302834/b302834.txt">Table of n, a(n) for n = 0..20000</a>
%H A302834 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A302834 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A302834 G.f.: (1/(1 - x))*Sum_{j>=0} x^(j^3)/Product_{k=1..j} (1 - x^(k^3)).
%F A302834 From _Vaclav Kotesovec_, Apr 13 2018: (Start)
%F A302834 a(n) ~ sqrt(3) * exp(4*(Gamma(1/3)*Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) / (8 * Pi^2 * sqrt(n)).
%F A302834 a(n) ~ 3^(3/2) * n^(3/4) / (Gamma(1/3)*Zeta(4/3))^(3/4) * A003108(n). (End)
%p A302834 b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
%p A302834       b(n, i-1)+ `if`(i^3>n, 0, b(n-i^3, i)))
%p A302834     end:
%p A302834 a:= n-> b(n, iroot(n, 3)):
%p A302834 seq(a(n), n=0..100);  # _Alois P. Heinz_, Apr 13 2018
%t A302834 nmax = 64; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k^3), {k, 1, nmax}], {x, 0, nmax}], x]
%t A302834 nmax = 64; CoefficientList[Series[1/(1 - x) Sum[x^j^3/Product[(1 - x^k^3), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]
%Y A302834 Cf. A000070, A000578, A003108, A078128, A298434, A302833.
%K A302834 nonn
%O A302834 0,2
%A A302834 _Ilya Gutkovskiy_, Apr 13 2018