A139215 Expansion of q^(-1) * psi(q) * phi(q^9) / (psi(q^3) * psi(q^6)) in power of q where phi(), psi() are Ramanujan theta functions.

1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, -3, 0, -4, 0, 0, 0, 4, 0, 5, 0, 0, 0, -7, 0, -8, 0, 0, 0, 12, 0, 14, 0, 0, 0, -17, 0, -20, 0, 0, 0, 24, 0, 28, 0, 0, 0, -36, 0, -40, 0, 0, 0, 52, 0, 56, 0, 0, 0, -71, 0, -80, 0, 0, 0, 96, 0, 109, 0, 0, 0, -133

Offset

-1,11

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 = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^2)^2 * eta(q^3) * eta(q^18)^5 / (eta(q) * eta(q^6) * eta(q^9)^2 * eta(q^12)^2 * eta(q^36)^2) in powers of q.

Euler transform of period 36 sequence [ 1, -1, 0, -1, 1, -1, 1, -1, 2, -1, 1, 1, 1, -1, 0, -1, 1, -4, 1, -1, 0, -1, 1, 1, 1, -1, 2, -1, 1, -1, 1, -1, 0, -1, 1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139213.

a(n) = -(-1)^n * A139216(n). a(2*n) = 0 unless n=0.

a(3*n + 1) = 0. a(6*n + 3) = - A217786(n). - Michael Somos, Sep 07 2015

Example

G.f. = 1/q + 1 - q^3 + 2*q^9 + 2*q^11 - 3*q^15 - 4*q^17 + 4*q^21 + 5*q^23 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 3, 0, q^9] / (EllipticTheta[ 2, 0, q^(3/2)] EllipticTheta[ 2, 0, q^3]), {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)^5 / (eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2 * eta(x^12 + A)^2 * eta(x^36 + A)^2), n))};

Crossrefs

Cf. A139216, A217786.

Sequence in context: A282695 A292936 A062590 * A139216 A348692 A355432

Adjacent sequences: A139212 A139213 A139214 * A139216 A139217 A139218

Keyword

sign

Author

Michael Somos, Apr 11 2008

Status

approved