A208850 Expansion of phi(q^2) / phi(-q) in powers of q where phi() is a Ramanujan theta function.
1, 2, 6, 12, 22, 40, 68, 112, 182, 286, 440, 668, 996, 1464, 2128, 3056, 4342, 6116, 8538, 11820, 16248, 22176, 30068, 40528, 54308, 72378, 95976, 126648, 166352, 217560, 283344, 367552, 474998, 611624, 784812, 1003712, 1279562, 1626216, 2060708, 2603856
Offset: 0
Keywords
Examples
1 + 2*q + 6*q^2 + 12*q^3 + 22*q^4 + 40*q^5 + 68*q^6 + 112*q^7 + 182*q^8 + ...
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
Crossrefs
Cf. A208589.
Programs
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Mathematica
nmax=60; CoefficientList[Series[Product[(1-x^(4*k))^5 / ((1-x^k)^2 * (1-x^(2*k)) * (1-x^(8*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *) a[n_] := SeriesCoefficient[EllipticTheta[3, 0, q^2]/EllipticTheta[3, 0, -q], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 27 2017 *) -
PARI
{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2), n))}
Formula
Expansion of eta(q^4)^5 / (eta(q)^2 * eta(q^2) * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ 2, 3, 2, -2, 2, 3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A208589.
G.f.: (Sum_k x^(2 * k^2)) / (Sum_k (-1)^k * x^k^2).
a(n) ~ exp(sqrt(n)*Pi)/(8*n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
Comments