A210067 Expansion of (phi(-q) / phi(q^2))^2 in powers of q where phi() is a Ramanujan theta function.

1, -4, 0, 16, 0, -56, 0, 160, 0, -404, 0, 944, 0, -2072, 0, 4320, 0, -8648, 0, 16720, 0, -31360, 0, 57312, 0, -102364, 0, 179104, 0, -307672, 0, 519808, 0, -864960, 0, 1419456, 0, -2299832, 0, 3682400, 0, -5831784, 0, 9141808, 0, -14194200, 0, 21842368, 0

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 - 4*q + 16*q^3 - 56*q^5 + 160*q^7 - 404*q^9 + 944*q^11 - 2072*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. A001938, A127393, A131126, A134746, A210030.

Programs

Mathematica

a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, q^2])^2, {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Nov 29 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)^2, n))}

Formula

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

Euler transform of period 8 sequence [ -4, -6, -4, 4, -4, -6, -4, 0, ...].

a(2*n) = 0 unless n=0. a(2*n + 1) = -4 * A001938(n) = -A127393(n).

a(n) = (-1)^n * A134746(n).

Convolution inverse of A131126. Convolution square of A210030.

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = -32 - 24*sqrt(2) + 4*sqrt(140+99*sqrt(2)). - Simon Plouffe, Mar 02 2021