A210066 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

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 ( (eta(q) / eta(q^8))^2 * (eta(q^4) / eta(q^2))^7)^2 in powers of q.

Euler transform of period 8 sequence [ -4, 10, -4, -4, -4, 10, -4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A134746.

a(n) = (-1)^n * A131126(n). Convolution inverse of A134746. Convolution square of A210065.

a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (2^(17/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017

Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 1/2 + sqrt(-4+3*sqrt(2)). - Simon Plouffe, Mar 02 2021

Example

1 - 4*q + 16*q^2 - 48*q^3 + 128*q^4 - 312*q^5 + 704*q^6 - 1504*q^7 + 3072*q^8 + ...

Mathematica

nmax = 40; CoefficientList[Series[Product[((1 - x^k) / (1 - x^(8*k)))^4 * (1 + x^(2*k))^14, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2017 *)
a[n_] := SeriesCoefficient[(EllipticTheta[3, 0, q^2]/ EllipticTheta[3, 0, q])^2, {q, 0, n}]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 29 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( (eta(x + A) / eta(x^8 + A))^2 * (eta(x^4 + A) / eta(x^2 + A))^7)^2, n))}

Crossrefs

Cf. A131126, A134746, A210065.

Sequence in context: A203248 A223839 A071009 * A131126 A159964 A058922

Adjacent sequences: A210063 A210064 A210065 * A210067 A210068 A210069

Keyword

sign

Author

Michael Somos, Mar 16 2012

Status

approved