A143068 Expansion of phi(q) / phi(-q^6) in powers of q where phi() is a Ramanujan theta function

1, 2, 0, 0, 2, 0, 2, 4, 0, 2, 4, 0, 4, 8, 0, 4, 10, 0, 8, 16, 0, 8, 20, 0, 14, 30, 0, 16, 36, 0, 24, 52, 0, 28, 64, 0, 42, 88, 0, 48, 108, 0, 68, 144, 0, 80, 176, 0, 108, 230, 0, 128, 280, 0, 170, 360, 0, 200, 436, 0, 260, 552, 0, 308, 666, 0, 392, 832, 0

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 eta(q^2)^5 * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^2 in powers of q.

Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -1, 2, -1, 2, -3, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (3/2)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143066.

a(3*n + 2) = 0.

G.f.: ( Sum_{k in Z} x^k^2 ) / ( Sum_{k in Z} (-x^6)^k^2 ).

Example

G.f. = 1 + 2*q + 2*q^4 + 2*q^6 + 4*q^7 + 2*q^9 + 4*q^10 + 4*q^12 + 8*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q^6], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};

Crossrefs

Cf. A143066.

Sequence in context: A260309 A046113 A262938 * A261202 A334596 A291900

Adjacent sequences: A143065 A143066 A143067 * A143069 A143070 A143071

Keyword

nonn

Author

Michael Somos, Jul 21 2008

Status

approved