A131126 Expansion of (phi(q^2) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function.

1, 4, 16, 48, 128, 312, 704, 1504, 3072, 6036, 11488, 21264, 38400, 67864, 117632, 200352, 335872, 554952, 904784, 1457136, 2320128, 3655296, 5702208, 8813472, 13504512, 20523996, 30952544, 46340832, 68901888, 101777112, 149403264, 218016640, 316342272

Offset: 0

Author

Michael Somos, Jun 15 2007

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 4*q + 16*q^2 + 48*q^3 + 128*q^4 + 312*q^5 + 704*q^6 + 1504*q^7 + ...
		

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. A014969, A107035.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[((1 - x^(4*k))^5 / ((1 - x^k)^2 * (1 - x^(2*k)) * (1 - x^(8*k))^2))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^2] / EllipticTheta[ 4, 0, q])^2, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)

PARI

{a(n) = my(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 ((phi(q) / phi(-q))^2 + 1) / 2 in powers of q where phi() is a Ramanujan theta function.

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

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

a(n) = 4 * A107035(n) unless n=0. 2 * a(n) = A014969(n) unless n=0.

a(n) ~ exp(sqrt(2*n)*Pi) / (16 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/2 + (1/8)*sqrt(8 + 6*sqrt(2)). - Simon Plouffe, Mar 04 2021