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A303904 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(k^3)).

%I A303904 #7 May 04 2018 07:11:13
%S A303904 1,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,
%T A303904 6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,10,10,
%U A303904 10,10,10,10,10,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13
%N A303904 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(k^3)).
%C A303904 Partial sums of A279329.
%H A303904 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A303904 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A303904 a(n) ~ exp(2^(7/4) * ((2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * 3^(5/4) / (2^(15/8) * sqrt(Pi) * ((2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3))^(3/8) * n^(1/8)). - _Vaclav Kotesovec_, May 04 2018
%p A303904 b:= proc(n, i) option remember; `if`(n<0, 0,
%p A303904      `if`(n=0, 1, `if`(n>i^2*(i+1)^2/4, 0, (t->
%p A303904        b(t, min(t, i-1)))(n-i^3)+b(n, i-1))))
%p A303904     end:
%p A303904 a:= proc(n) option remember; `if`(n<0, 0,
%p A303904        b(n, iroot(n, 3))+a(n-1))
%p A303904     end:
%p A303904 seq(a(n), n=0..100);  # _Alois P. Heinz_, May 02 2018
%t A303904 nmax = 91; CoefficientList[Series[1/(1 - x) Product[1 + x^k^3, {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x]
%Y A303904 Cf. A000578, A003997, A036469, A038348, A248801, A279329, A302834.
%K A303904 nonn
%O A303904 0,2
%A A303904 _Ilya Gutkovskiy_, May 02 2018