A164273 Expansion of phi(-q) * phi(q^3) in powers of q where phi() is a Ramanujan theta function.

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

Offset: 0

Author

Michael Somos, Aug 11 2009

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^7 - 2*q^9 - 2*q^12 - 4*q^13 + 6*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. A164272.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0 ,q^3], {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164273[n_] := SeriesCoefficient[f[-q, -q]*f[q^3, q^3], {q, 0, n}]; Table[A164273[n], {n, 0, 50}] (* G. C. Greubel, Sep 16 2017 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

Formula

Expansion of eta(q)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ -2, -1, 0, -1, -2, -4, -2, -1, 0, -1, -2, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112605.

a(n) = (-1)^n * A164272(n).