A226350 Expansion of psi(x) * psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

1, 1, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -2, 0

Offset

0,10

Comments

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

Links

Table of n, a(n) for n=0..82.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^6)) in powers of q.

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

a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = 0.

Example

1 + x - x^4 - 2*x^9 - x^12 - x^13 + 2*x^21 - x^24 + 2*x^28 + 2*x^33 + ...
q + q^3 - q^9 - 2*q^19 - q^25 - q^27 + 2*q^43 - q^49 + 2*q^57 + 2*q^67 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, Pi/4, q^3]/8^(1/2), {q, 0, 2 n + 1}]

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)), n))}

Crossrefs

Sequence in context: A354105 A370366 A342419 * A112609 A134363 A054015

Adjacent sequences: A226347 A226348 A226349 * A226351 A226352 A226353

Keyword

sign

Author

Michael Somos, Jun 04 2013

Status

approved