A143068 Expansion of phi(q) / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, 0, 2, 0, 2, 4, 0, 2, 4, 0, 4, 8, 0, 4, 10, 0, 8, 16, 0, 8, 20, 0, 14, 30, 0, 16, 36, 0, 24, 52, 0, 28, 64, 0, 42, 88, 0, 48, 108, 0, 68, 144, 0, 80, 176, 0, 108, 230, 0, 128, 280, 0, 170, 360, 0, 200, 436, 0, 260, 552, 0, 308, 666, 0, 392, 832, 0

Offset: 0

Author

Michael Somos, Jul 21 2008

Keywords

• nonn

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^4 + 2*q^6 + 4*q^7 + 2*q^9 + 4*q^10 + 4*q^12 + 8*q^13 + ...
		

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. A143066.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q^6], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};

Formula

Expansion of eta(q^2)^5 * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^2 in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = (3/2)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143066.

a(3*n + 2) = 0.

G.f.: ( Sum_{k in Z} x^k^2 ) / ( Sum_{k in Z} (-x^6)^k^2 ).