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

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

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

Example

1 + 2*q + 6*q^2 + 12*q^3 + 22*q^4 + 40*q^5 + 68*q^6 + 112*q^7 + 182*q^8 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A208589.

Sequence in context: A182977 A116658 A210065 * A131520 A086953 A101953

Adjacent sequences: A208847 A208848 A208849 * A208851 A208852 A208853

Keyword

nonn

Author

Michael Somos, Mar 02 2012

Status

approved