A217771 Expansion of (phi(-x) / phi(-x^3))^2 in powers of x where phi() is a Ramanujan theta function.

1, -4, 4, 4, -12, 8, 12, -32, 20, 28, -72, 48, 60, -152, 96, 120, -300, 184, 228, -560, 344, 416, -1008, 608, 732, -1756, 1048, 1252, -2976, 1768, 2088, -4928, 2900, 3408, -7992, 4672, 5460, -12728, 7408, 8600, -19944, 11544, 13344, -30800, 17744, 20424

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)^4 * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)^4) in powers of q.

Euler transform of period 6 sequence [ -4, -2, 0, -2, -4, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u) * (u + v^2) - 4 * u.

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (3 + u * v)^2 - v * (3*u + v)^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A217786.

a(n) = - 4 * A123649(n) unless n=0.

Convolution inverse of A186924. Convolution square of A139137.

Example

G.f. = 1 - 4*x + 4*x^2 + 4*x^3 - 12*x^4 + 8*x^5 + 12*x^6 - 32*x^7 + 20*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 / EllipticTheta[ 4, 0, q^3]^2, {q, 0, n}]; (* Michael Somos, Mar 24 2013 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)^4), n))}

Crossrefs

Cf. A123649, A186924, A217786.

Sequence in context: A151896 A267191 A170897 * A261321 A245517 A179526

Adjacent sequences: A217768 A217769 A217770 * A217772 A217773 A217774

Keyword

sign

Author

Michael Somos, Mar 24 2013

Status

approved