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

0,2

Comments

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

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

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

Example

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

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 *)

Prog

(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))};

Crossrefs

Cf. A014969, A107035.

Sequence in context: A223839 A071009 A210066 * A159964 A058922 A215723

Adjacent sequences: A131123 A131124 A131125 * A131127 A131128 A131129

Keyword

nonn

Author

Michael Somos, Jun 15 2007

Status

approved