A252706 Expansion of phi(-q) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

1, -2, 0, 2, -2, 0, 4, -4, 0, 6, -8, 0, 10, -12, 0, 16, -18, 0, 24, -28, 0, 36, -40, 0, 52, -58, 0, 74, -84, 0, 104, -116, 0, 144, -160, 0, 198, -220, 0, 268, -296, 0, 360, -396, 0, 480, -528, 0, 634, -694, 0, 832, -908, 0, 1084, -1184, 0, 1404, -1528, 0

Offset: 0

Author

Michael Somos, Apr 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^3 - 2*q^4 + 4*q^6 - 4*q^7 + 6*q^9 - 8*q^10 + 10*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. A097197, A098151, A101195, A139137.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];

PARI

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

Formula

Expansion of f(-q, -q^2) / f(q, q^2) in powers of q where f(,) is Ramanujan's two-variable theta function.

Euler transform of period 6 sequence [ -2, -1, 0, -1, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A101195.

G.f.: Product_{k>0} (1 - x^k + x^(2*k)) / (1 + x^k + x^(2*k)).

a(n) = (-1)^n * A139137(n).

Convolution inverse is A098151.

a(3*n + 2) = 0. a(3*n) = A098151(n). a(3*n + 1) = -2 * A097197(n).