A210065 Expansion of phi(q^2) / phi(q) in powers of q where phi() is a Ramanujan theta function.

1, -2, 6, -12, 22, -40, 68, -112, 182, -286, 440, -668, 996, -1464, 2128, -3056, 4342, -6116, 8538, -11820, 16248, -22176, 30068, -40528, 54308, -72378, 95976, -126648, 166352, -217560, 283344, -367552, 474998, -611624, 784812, -1003712, 1279562, -1626216

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

			G.f. = 1 - 2*q + 6*q^2 - 12*q^3 + 22*q^4 - 40*q^5 + 68*q^6 - 112*q^7 + 182*q^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A080015, A208850.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[((1 - x^k) / (1 - x^(8*k)))^2 * (1 + x^(2*k))^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 17 2017 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]/ eta[q^8])^2*(eta[q^4]/eta[q^2])^7, {q, 0, 50}], q] (* G. C. Greubel, Aug 11 2018 *)

PARI

{a(n) = my(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, n))};

Formula

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

Euler transform of period 8 sequence [-2, 5, -2, -2, -2, 5, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A080015.

a(n) = (-1)^n * A208850(n). Convolution inverse of A080015.

a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (8*n^(3/4)). - Vaclav Kotesovec, Nov 17 2017