A263145 - OEIS

%I #8 Oct 12 2015 10:32:12

%S 1,0,0,0,1,0,0,0,0,2,0,0,0,2,3,0,0,0,4,4,0,0,1,10,5,0,0,6,16,6,0,0,14,

%T 28,7,0,3,32,40,8,0,10,63,60,9,0,33,112,80,10,3,74,187,110,11,14,161,

%U 300,140,12,46,308,455,182,14,120,568,672,224,26,283

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

%H Vaclav Kotesovec, <a href="/A263145/b263145.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2).

%F a(n) ~ 2^(27/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(32400*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (900*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 3^(4/3) * 2^(2/3) * 5^(1/3) * n^(2/3) / 20) / (30 * sqrt(Pi) * n^(2/3)).

%t nmax = 100; CoefficientList[Series[Product[(1+x^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A263146, A263147, A263148, A263141, A263140, A262878, A263138.

%K nonn

%O 0,10

%A _Vaclav Kotesovec_, Oct 10 2015