A216060 Expansion of (phi(q) / phi(q^4))^2 in powers of q where phi() is a Ramanujan theta function.

1, 4, 4, 0, 0, -8, -16, 0, 0, 20, 56, 0, 0, -40, -160, 0, 0, 72, 404, 0, 0, -128, -944, 0, 0, 220, 2072, 0, 0, -360, -4320, 0, 0, 576, 8648, 0, 0, -904, -16720, 0, 0, 1384, 31360, 0, 0, -2088, -57312, 0, 0, 3108, 102364, 0, 0, -4552, -179104, 0, 0, 6592

Offset: 0

Author

Michael Somos, Aug 31 2012

Keywords

• sign

Comments

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

Examples

			1 + 4*q + 4*q^2 - 8*q^5 - 16*q^6 + 20*q^9 + 56*q^10 - 40*q^13 - 160*q^14 + ...
		

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, A079006, A208274.

Programs

Mathematica

a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]/EllipticTheta[3, 0, q^4])^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 04 2017 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^8 + A)^5))^2, n))}

Formula

Expansion of (eta(q^2)^5 * eta(q^16)^2 / (eta(q)^2 * eta(q^8)^5))^2 in powers of q.

Euler transform of period 16 sequence [ 4, -6, 4, -6, 4, -6, 4, 4, 4, -6, 4, -6, 4, -6, 4, 0, ...].

a(4*n) = 0 unless n=0. a(4*n + 3) = 0. a(4*n + 1) = 4 * A079006(n). a(4*n + 2) = 4 * A001938(n).

Convolution square of A208274.

Empirical: Sum{n>=0} a(n)/exp(Pi*n) = 40 + 28*sqrt(2) - 8*sqrt(48+34*sqrt(2)). - Simon Plouffe, Mar 02 2021