A262938 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

Offset: 0

Author

Michael Somos, Oct 04 2015

Keywords

• sign

Comments

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

Examples

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

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

Crossrefs

Cf. A143068, A260256, A261877, A262160.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^6], {q, 0, n}];

PARI

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

Formula

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 6^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A262160.

a(n) = (-1)^n * A143068(n). a(2*n) = A260256(n). a(2*n + 1) = -2 * A261877(n).

a(3*n) = 0.