OFFSET
0,5
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 20 sequence [0, 1, 0, 1, 2, 1, 0, 1, 0, -2, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, ...].
Expansion of phi(q^5) / f(-q^2) in powers of q where phi(), f() are Ramanujan theta functions.
Expansion of G(x) * G(x^4) - x * H(x) * H(x^4) where G() = g.f. A003114 and H() = g.f. A003106 are the Rogers-Ramanujan functions.
G.f.: (1 + 2 * Sum_{k>0} x^(5*k^2)) / (Product_{k>0} (1 - x^(2*k))).
a(n) ~ exp(Pi*sqrt(n/3)) / (2*3^(1/4)*sqrt(10)*n^(3/4)). - Vaclav Kotesovec, Jul 11 2016
EXAMPLE
1/q + q^23 + 2*q^47 + 2*q^59 + 3*q^71 + 2*q^83 + 5*q^95 + 4*q^107 + ...
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1-x^(10*k))^5 / ( (1-x^(2*k)) * (1-x^(5*k))^2 * (1-x^(20*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(1/12)* eta[q^10]^5/(eta[q^2]*eta[q^5]^2*eta[q^20]^2), {q, 0, 50}], q] (* G. C. Greubel, Apr 17 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A)^5 / eta(x^2 + A) / eta(x^5 + A)^2 / eta(x^20 + A)^2, n))}
(PARI) q='q+O('q^99); Vec(eta(q^10)^5/(eta(q^2)*eta(q^5)^2*eta(q^20)^2)) \\ Altug Alkan, Apr 18 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 24 2005
STATUS
approved