OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..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
Expansion of (eta(q^2) * eta(q^5) * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10)))^2 in powers of q.
Euler transform of period 20 sequence [ 2, 0, 2, 2, 0, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 2, 2, 0, 2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 + u) * (1 + 5*u) * v * (1 + v) * (1 + 5*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. of A145740.
G.f.: x * (Product_{k>0} P(5, x^k) * P(20, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
Empirical: Sum_{n>=1} a(n)/exp(Pi*n) = -2/5 + (1/5)*sqrt(5). - Simon Plouffe, Mar 02 2021
EXAMPLE
q + 2*q^2 + 3*q^3 + 6*q^4 + 11*q^5 + 16*q^6 + 24*q^7 + 38*q^8 + 57*q^9 + ...
MATHEMATICA
nmax=60; CoefficientList[Series[Product[((1+x^k) * (1-x^(5*k)) * (1+x^(10*k)) / (1-x^(4*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, I*q^(5/2)]/EllipticTheta[ 2, 0, I*Sqrt[q]])^2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Dec 07 2017 *)
PROG
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)))^2, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 21 2013
STATUS
approved