A138270 Expansion of phi(-q^3) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

1, 0, 0, -2, -2, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, -2, 4, 0, 0, 4, 0, 0, 0, 0, -2, 0, 0, 4, 0, 0, 0, -4, 0, 0, 0, 0, -2, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, -2, 4, 0, 0, 4, 0, 0, 0

Offset

0,4

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 (eta(q^3) * eta(q^4))^2 / (eta(q^6) * eta(q^8)) in powers of q.

Euler transform of period 24 sequence [ 0, 0, -2, -2, 0, -1, 0, -1, -2, 0, 0, -3, 0, 0, -2, -1, 0, -1, 0, -2, -2, 0, 0, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 192^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112609.

a(3*n + 2) = a(4*n + 1) = a(4*n + 2) = 0.

a(4*n) = A164273(n). - Michael Somos, Sep 27 2015

Example

G.f. = 1 - 2*q^3 - 2*q^4 + 4*q^7 + 2*q^12 - 2*q^16 - 4*q^19 - 2*q^27 + 4*q^28 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* Michael Somos, Sep 27 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^4 + A))^2 / (eta(x^6 + A) * eta(x^8 + A)), n))};

Crossrefs

Cf. A112609, A164273.

Sequence in context: A355664 A254040 A062275 * A317643 A179011 A300465

Adjacent sequences: A138267 A138268 A138269 * A138271 A138272 A138273

Keyword

sign

Author

Michael Somos, Mar 10 2008, Apr 04 2008

Status

approved