A257400 Expansion of psi(q) * phi(q^2) * chi(-q^3) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

1, 1, 2, 2, -1, 0, -2, 0, 2, 0, 0, -4, -2, 0, 0, 0, -1, -4, 0, 2, 0, 0, -2, 0, -2, 1, 0, -2, 0, 0, 0, 0, 2, 4, -2, 0, 0, 0, 4, 0, 0, -4, 0, 2, 4, 0, 0, 0, -2, 1, 2, 4, 0, 0, 2, 0, 0, 4, 0, -4, 0, 0, 0, 0, -1, 0, -4, 2, 4, 0, 0, 0, 0, 2, 0, 2, -2, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Apr 21 2015

Keywords

• sign

Comments

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

Examples

			G.f. = 1 + q + 2*q^2 + 2*q^3 - q^4 - 2*q^6 + 2*q^8 - 4*q^11 - 2*q^12 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A257399.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 3, 0, q^2] QPochhammer[ q^3, q^6] / (2 q^(1/8)), {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A)^5 / (eta(x + A) * eta(x^8 + A)^2 * eta(x^6 + A)), n))};

Formula

Expansion of eta(q^3) * eta(q^4)^5 / (eta(q) * eta(q^8)^2 * eta(q^6)) in powers of q.

Euler transform of period 24 sequence [1, 1, 0, -4, 1, 1, 1, -2, 0, 1, 1, -4, 1, 1, 0, -2, 1, 1, 1, -4, 0, 1, 1, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2592^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A257399.

a(8*n + 7) = 0.