A261735 - OEIS

%I #15 Jul 29 2018 20:50:14

%S 1,-1,0,-1,1,-1,1,-1,3,-3,2,-3,4,-4,4,-5,8,-8,7,-9,11,-12,12,-14,20,

%T -21,19,-24,28,-30,31,-35,45,-48,47,-55,64,-68,71,-80,97,-103,104,

%U -119,135,-145,152,-168,198,-211,216,-243,272,-291,307,-337,386,-412

%N Expansion of Product_{k>=1} (1 + x^(8*k))/(1 + x^k).

%C In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd.

%H Seiichi Manyama, <a href="/A261735/b261735.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, p. 14.

%F a(n) ~ (-1)^n * exp(sqrt(5*n/6)*Pi/2) * 5^(1/4) / (2^(11/4)*3^(1/4)*n^(3/4)).

%p seq(coeff(series(mul((1+x^(8*k))/(1+x^k),k=1..n), x,n+1),x,n),n=0..60); # _Muniru A Asiru_, Jul 29 2018

%t nmax = 100; CoefficientList[Series[Product[(1 + x^(8*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261733 (m=9), A145707 (m=10).

%K sign

%O 0,9

%A _Vaclav Kotesovec_, Aug 30 2015