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Expansion of Product_{k>=1} (1 + k^3*x^k).
6

%I #11 Sep 07 2023 15:52:49

%S 1,1,8,35,91,405,1069,3799,8686,36744,86310,235776,686329,1605779,

%T 5230579,13191702,30608501,73907925,206052723,433747560,1324608945,

%U 2995740974,6973434054,15364943439,35816669079,86662644756,184871083828,502089539734,1098571699830

%N Expansion of Product_{k>=1} (1 + k^3*x^k).

%H Vaclav Kotesovec, <a href="/A265840/b265840.txt">Table of n, a(n) for n = 0..2000</a>

%F G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(3*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 14 2018

%F Conjecture: log(a(n)) ~ 3*sqrt(n/2) * (log(2*n) - 2). - _Vaclav Kotesovec_, Dec 27 2020

%t nmax = 40; CoefficientList[Series[Product[1 + k^3*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A022629, A092484, A265837, A265841, A265842.

%Y Column k=3 of A292189.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Dec 16 2015