A233458 Expansion of q^(-1) * (phi(q^2) * phi(-q) / psi(-q^2)^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.

1, -4, 12, -32, 66, -128, 232, -384, 639, -1024, 1596, -2496, 3774, -5632, 8328, -12032, 17283, -24576, 34520, -48288, 66882, -91904, 125568, -170112, 229244, -307200, 409236, -542912, 716412, -941056, 1231048, -1602816, 2079237, -2686976, 3459264, -4439616

Offset

-1,2

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 = -1..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1) * (phi(-q^4)^2 / (phi(q) * psi(q^4)))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of (eta(q)^2 * eta(q^4)^7 / (eta(q^2)^5 * eta(q^8)^4))^2 in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/8) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A232772.

a(2*n) = -4 * A014969(n). a(2*n - 1) = A112142(n).

Example

G.f. = 1/q - 4 + 12*q - 32*q^2 + 66*q^3 - 128*q^4 + 232*q^5 - 384*q^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (2 EllipticTheta[ 4, 0, q^4]^2 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^2]))^2, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (2 EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2] / EllipticTheta[ 2, Pi/4, q]^2)^2, {q, 0, n}]

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A)^7 / (eta(x^2 + A)^5 * eta(x^8 + A)^4))^2, n))}

Crossrefs

Cf. A014969, A112142, A232772.

Sequence in context: A028921 A028922 A066150 * A335687 A348864 A133212

Adjacent sequences: A233455 A233456 A233457 * A233459 A233460 A233461

Keyword

sign

Author

Michael Somos, Dec 10 2013

Status

approved