A224821 Expansion of theta_4(q)^2 * theta_4(q^3) in powers of q.

1, -4, 4, -2, 12, -16, 0, -8, 20, -4, 8, -8, 10, -32, 8, 0, 28, -24, 4, -8, 32, -16, 16, -16, 0, -28, 8, -2, 40, -48, 8, -8, 52, 0, 8, -16, 12, -64, 16, -8, 40, -24, 0, -24, 40, -16, 16, -16, 26, -28, 20, 0, 64, -80, 0, -16, 40, -24, 24, -8, 0, -64, 24, -8

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

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

G.f.: theta_4(q)^2 * theta_4(q^3) = (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(3*k^2)).

a(n) = (-1)^n * A034933(n). a(2*n) = A014458(n). a(9*n) = a(n). a(9*n + 6) = 0.

Example

1 - 4*q + 4*q^2 - 2*q^3 + 12*q^4 - 16*q^5 - 8*q^7 + 20*q^8 - 4*q^9 + ...

Mathematica

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

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^3 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))}
(PARI) q='q+O('q^99); Vec(eta(q)^4*eta(q^3)^2/(eta(q^2)^2*eta(q^6))) \\ Altug Alkan, Apr 12 2018

Crossrefs

Cf. A014458, A034933.

Sequence in context: A202322 A365797 A232523 * A034933 A320148 A320147

Adjacent sequences: A224818 A224819 A224820 * A224822 A224823 A224824

Keyword

sign

Author

Michael Somos, Jul 20 2013

Status

approved