A229723 Expansion of psi(q) * chi(-q^3) * phi(-q^6) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

1, 1, 0, 0, -1, 0, -2, -2, 0, -2, 2, 0, 0, 0, 0, 4, -1, 0, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, -2, 0, 0, -2, 0, -4, 0, 0, 2, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, -2, -4, 0, 0, 2, 0, 4, 0, 0, 4, -1, 0, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, -2, 0, -2

Offset: 0

Author

Michael Somos, Sep 27 2013

Keywords

• sign

Comments

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

Examples

			G.f. = 1 + q - q^4 - 2*q^6 - 2*q^7 - 2*q^9 + 2*q^10 + 4*q^15 - q^16 + ...
		

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. A128583.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^3, q^6] EllipticTheta[ 4, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^3] QPochhammer[ q^6]/ (QPochhammer[ q] QPochhammer[ q^12]), {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^12 + A)), n))};

Formula

Expansion eta(q^2)^2 * eta(q^3) * eta(q^6) / (eta(q) * eta(q^12)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 13824^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128583.

a(3*n + 2) = 0.