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Expansion of Product_{k>=1} (1 + k^5*x^k).
6

%I #11 Sep 07 2023 15:56:33

%S 1,1,32,275,1267,11925,51445,406183,1406614,14690040,51144366,

%T 251885088,1481359033,5108404955,42614629915,158222158038,

%U 588574803125,2360755022421,13255325882835,39266011999104,325719196861377,1031732678138822,3791401325667894

%N Expansion of Product_{k>=1} (1 + k^5*x^k).

%H Vaclav Kotesovec, <a href="/A265842/b265842.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^(5*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Oct 18 2018

%F Conjecture: log(a(n)) ~ 5*sqrt(n/2) * (log(2*n) - 2). - _Vaclav Kotesovec_, Dec 27 2020

%t nmax = 40; CoefficientList[Series[Product[1 + k^5*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A022629, A092484, A265839, A265840, A265841.

%Y Column k=5 of A292189.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Dec 16 2015