cp's OEIS Frontend

A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.

%I A263140 #10 Sep 08 2017 06:49:38
%S A263140 1,1,0,2,2,3,4,5,10,11,16,20,31,39,50,71,93,124,154,211,271,357,449,
%T A263140 587,762,968,1233,1571,2021,2535,3220,4049,5145,6431,8070,10105,12670,
%U A263140 15784,19619,24447,30348,37635,46464,57532,70945,87477,107456,132192,162220
%N A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.
%H A263140 Vaclav Kotesovec, <a href="/A263140/b263140.txt">Table of n, a(n) for n = 0..10000</a>
%H A263140 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015
%F A263140 G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^j/(1 - x^(2*j))^2).
%F A263140 a(n) ~ exp(-Pi^4 / (5184*Zeta(3)) + Pi^2 * n^(1/3) / (8 * 3^(4/3) * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)/4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3)* sqrt(Pi) * n^(2/3)).
%t A263140 nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A263140 nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]
%Y A263140 Cf. A035528, A263149, A263150, A263199, A262878, A263138, A263145, A292037.
%K A263140 nonn
%O A263140 0,4
%A A263140 _Vaclav Kotesovec_, Oct 10 2015