A210030 Expansion of phi(-q) / phi(q^2) in powers of q where phi() is a Ramanujan theta function.

1, -2, -2, 4, 6, -8, -12, 16, 22, -30, -40, 52, 68, -88, -112, 144, 182, -228, -286, 356, 440, -544, -668, 816, 996, -1210, -1464, 1768, 2128, -2552, -3056, 3648, 4342, -5160, -6116, 7232, 8538, -10056, -11820, 13872, 16248, -18996, -22176, 25844, 30068

Offset: 0

Author

Michael Somos, Mar 16 2012

Keywords

• sign

Comments

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

Examples

			1 - 2*q - 2*q^2 + 4*q^3 + 6*q^4 - 8*q^5 - 12*q^6 + 16*q^7 + 22*q^8 + ...
		

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. A080015, A080054, A208850.

Programs

Mathematica

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 17 2017 *)

PARI

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

Formula

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

Euler transform of period 8 sequence [ -2, -3, -2, 2, -2, -3, -2, 0, ...].

G.f.: (Sum_k (-1)^k * x^k^2) / (Sum_k x^(2 * k^2)).

a(n) = (-1)^n * A080015(n) = (-1)^[(n + 1) / 4] * A080054(n).

Convolution inverse of A208850.