A232392 Expansion of q^(-1) * phi(q^2)^2 / (phi(q) * psi(q^8)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, -2, 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

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^4)^12 / (eta(q^2)^9 * eta(q^8)^3 * eta(q^16)^2) in powers of q.

Euler transform of period 16 sequence [ -2, 7, -2, -5, -2, 7, -2, -2, -2, 7, -2, -5, -2, 7, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A212318.

a(n) = -(-1)^n * A176143(n). a(2*n) = -2 * A014969(n).

Apparently a(n) = A215346(n) for n <> 0. - R. J. Mathar, Nov 27 2013

Example

G.f. = 1/q - 2 + 8*q - 16*q^2 + 34*q^3 - 64*q^4 + 112*q^5 - 192*q^6 + ...

Mathematica

a[ n_]:= SeriesCoefficient[2*EllipticTheta[3, 0, q^2]^2/(EllipticTheta[3, 0, q]*EllipticTheta[2, 0, q^4]), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* modified by G. C. Greubel, Mar 14 2018 *)

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)^12 / (eta(x^2 + A)^9 * eta(x^8 + A)^3 * eta(x^16 + A)^2), n))};
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^4)^12/(eta(q^2)^9*eta(q^8)^3*eta(q^16)^2)) \\ Altug Alkan, Mar 20 2018

Crossrefs

Cf. A014969, A176143, A212318, A215346.

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

Sequence in context: A232358 A212318 A346461 * A176143 A296946 A096227

Adjacent sequences: A232389 A232390 A232391 * A232393 A232394 A232395

Keyword

sign

Author

Michael Somos, Nov 23 2013

Extensions

Offset corrected by Altug Alkan, Mar 22 2018

Status

approved