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A208850 Expansion of phi(q^2) / phi(-q) in powers of q where phi() is a Ramanujan theta function. 7
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 11 06:11 EDT 2021. Contains 342886 sequences. (Running on oeis4.)