A262966 Expansion of phi(-q^3) / phi(-q^2) in powers of q where phi() is a Ramanujan theta function.

1, 0, 2, -2, 4, -4, 8, -8, 14, -16, 24, -28, 42, -48, 68, -80, 108, -128, 170, -200, 260, -308, 392, -464, 584, -688, 856, -1010, 1240, -1460, 1780, -2088, 2526, -2960, 3552, -4152, 4956, -5776, 6856, -7976, 9416, -10928, 12848, -14872, 17412, -20116, 23456

Offset

0,3

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..2500

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^3)^2 * eta(q^4) / (eta(q^2)^2 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [0, 2, -2, 1, 0, 1, 0, 1, -2, 2, 0, 0, ...].

a(n) ~ (-1)^n * exp(sqrt(n/2)*Pi) / (2^(9/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015

Example

G.f. = 1 + 2*q^2 - 2*q^3 + 4*q^4 - 4*q^5 + 8*q^6 - 8*q^7 + 14*q^8 - 16*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] / EllipticTheta[ 4, 0, q^2], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q^3)^2*eta(q^4)/(eta(q^2)^2*eta(q^6))) \\ Altug Alkan, Jul 31 2018

Crossrefs

Sequence in context: A145810 A172148 A205478 * A034397 A200750 A344607

Adjacent sequences: A262963 A262964 A262965 * A262967 A262968 A262969

Keyword

sign

Author

Michael Somos, Oct 05 2015

Status

approved