login
Expansion of Product_{k>=1} (1 + k^4*x^k).
6

%I #11 Sep 07 2023 15:56:53

%S 1,1,16,97,337,2177,7313,38529,108594,717186,2053522,7527458,30757155,

%T 88042387,448973459,1390503396,4087546309,12699966117,49599776261,

%U 124699632310,608410782855,1651128186296,4862631132392,13170300313769,39285370060347,130999461143020

%N Expansion of Product_{k>=1} (1 + k^4*x^k).

%H Vaclav Kotesovec, <a href="/A265841/b265841.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^(4*k)*x^(j*k)/k). - _Ilya Gutkovskiy_, Jun 14 2018

%F Conjecture: log(a(n)) ~ 4*sqrt(n/2) * (log(2*n) - 2). - _Vaclav Kotesovec_, Dec 27 2020

%t nmax = 40; CoefficientList[Series[Product[1 + k^4*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A022629, A092484, A265838, A265840, A265842.

%Y Column k=4 of A292189.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Dec 16 2015