A260110 Expansion of f(-x, -x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

1, -2, 0, 0, 3, -2, 0, 0, 3, -4, 0, 0, 2, -2, 0, 0, 2, -2, 0, 0, 3, -2, 0, 0, 4, -2, 0, 0, 1, -6, 0, 0, 2, -2, 0, 0, 4, -2, 0, 0, 2, 0, 0, 0, 4, -2, 0, 0, 1, -4, 0, 0, 2, -4, 0, 0, 2, -4, 0, 0, 1, -2, 0, 0, 8, 0, 0, 0, 2, -4, 0, 0, 2, -2, 0, 0, 2, -2, 0, 0, 0

Offset: 0

Author

Michael Somos, Jul 16 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*x + 3*x^4 - 2*x^5 + 3*x^8 - 4*x^9 + 2*x^12 - 2*x^13 + 2*x^16 + ...
G.f. = q - 2*q^7 + 3*q^25 - 2*q^31 + 3*q^49 - 4*q^55 + 2*q^73 - 2*q^79 + ...
		

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. A134177, A190615, A229723, A260089.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x^4, x^8], {x, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x^2 + A) * eta(x^4 + A) * eta(x^24 + A)), n))};

Formula

Expansion of q^(-1/6) * eta(q)^2 * eta(q^8) * eta(q^12)^2 / (eta(q^2) * eta(q^4) * eta(q^24)) in powers of q.

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

a(n) = A134177(3*n) = A190615(3*n) = A229723(6*n + 1). a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A113780(n). a(4*n + 1) = -2 * A260089(n).