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], Sep 30 2015
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2) * eta(q^3) * eta(q^12) * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9)^2 * eta(q^36)^2) in powers of q.
Euler transform of period 36 sequence [ 1, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, -2, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, ...].
a(n) = A132975(n) unless n=0.
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2 * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
EXAMPLE
1 + q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1+x^(6*k)) * (1+x^(9*k))^5 * (1-x^(9*k))^3 / ((1-x^(4*k)) * (1+x^(3*k)) * (1-x^(36*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 07 2012
STATUS
approved