A215346 Expansion of (1/q) * phi(-q) * phi(q^4) / (phi(q) * psi(q^8)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, -4, 8, -16, 34, -64, 112, -192, 319, -512, 808, -1248, 1886, -2816, 4144, -6016, 8643, -12288, 17296, -24144, 33442, -45952, 62720, -85056, 114620, -153600, 204728, -271456, 358204, -470528, 615344, -801408, 1039621, -1343488, 1729920, -2219808, 2838920

Offset

-1,2

Comments

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

A058516, A176143, A214035, A215346 are all essentially the same sequence. - N. J. A. Sloane, Aug 08 2012

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of ( eta(q)^2 * eta(q^8)^3 / (eta(q^2)^3 * eta(q^16)^2))^2 in powers of q.

Euler transform of period 16 sequence [ -4, 2, -4, 2, -4, 2, -4, -4, -4, 2, -4, 2, -4, 2, -4, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4 * g(t) where q = exp(2 Pi i t) and g() is g.f. for A215348.

a(n) = (-1)^n * A214035(n). a(2*n) = -4 * A131126(n). Convolution inverse of A215348.

Example

1/q - 4 + 8*q - 16*q^2 + 34*q^3 - 64*q^4 + 112*q^5 - 192*q^6 + 319*q^7 + ...

Mathematica

QP = QPochhammer; s = (QP[q]^2*(QP[q^8]^3/(QP[q^2]^3*QP[q^16]^2)))^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( eta(x + A)^2 * eta(x^8 + A)^3 / (eta(x^2 + A)^3 * eta(x^16 + A)^2))^2, n))}

Crossrefs

Cf. A000122, A000700, A010054, A121373, A131126, A214035, A215348.

Cf. A058516, A176143, A214035, A215346.

Sequence in context: A019458 A264166 A330913 * A214035 A040014 A141069

Adjacent sequences: A215343 A215344 A215345 * A215347 A215348 A215349

Keyword

sign

Author

Michael Somos, Aug 08 2012

Status

approved