A112128 Expansion of phi(q^4) / phi(q) in powers of q where phi() is a Ramanujan theta function.

1, -2, 4, -8, 16, -28, 48, -80, 128, -202, 312, -472, 704, -1036, 1504, -2160, 3072, -4324, 6036, -8360, 11488, -15680, 21264, -28656, 38400, -51182, 67864, -89552, 117632, -153836, 200352, -259904, 335872, -432480, 554952, -709728, 904784, -1149916, 1457136

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

C. Adiga and N. Anitha, A note on a continued fraction of Ramanujan, Bull. Austral. Math. Soc. 70 (2004), pp. 489-497. MR2103981 (2005g:11009)

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q) / eta(q^16))^2 * (eta(q^8) / eta(q^2))^5 in powers of q.

Euler transform of period 16 sequence [ -2, 3, -2, 3, -2, 3, -2, -2, -2, 3, -2, 3, -2, 3, -2, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - (1 - 2*u + 2*u^2) * (1 - 2*v + 2*v^2).

G.f.: (Sum_{k in Z} x^(4*k^2)) / (Sum_{k in Z} x^(k^2)) = theta_3(0, x^4) / theta_3(0, x).

G.f.: Product_{k>0} ((1 + x^(2*k)) * (1 + x^(4*k)))^3 / ((1 + x^k) * (1 + x^(8*k)))^2.

Expansion of continued fraction 1 / (1 + 2*x / (1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...)))).

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 1/2 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208724.

(-1)^n * a(n) = A208933(n). a(2*n) = A131126(n). a(2*n + 2) = -2 * A093160(n). - Michael Somos, Dec 11 2016

Convolution inverse of A208274. - Michael Somos, Dec 11 2016

a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (2^(7/2) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Example

G.f. = 1 - 2*q + 4*q^2 - 8*q^3 + 16*q^4 - 28*q^5 + 48*q^6 - 80*q^7 + 128*q^8 + ...

Mathematica

QP = QPochhammer; s = QP[q]^2*(QP[q^8]^5/QP[q^2]^5/QP[q^16]^2) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^4] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Dec 11 2016 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A)^5 / (eta(x^2 + A)^5 * eta(x^16 + A)^2), n))};

Crossrefs

Cf. A093160, A131126, A208724, A208933.

Sequence in context: A089055 A305122 A276677 * A208933 A227036 A172020

Adjacent sequences: A112125 A112126 A112127 * A112129 A112130 A112131

Keyword

sign

Author

Michael Somos, Aug 27 2005

Status

approved