A215349 Expansion of q * phi(-q) * psi(q^8) / (phi(q) * phi(q^4)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, -4, 8, -16, 30, -48, 80, -128, 197, -312, 472, -704, 1046, -1504, 2160, -3072, 4306, -6036, 8360, -11488, 15712, -21264, 28656, -38400, 51127, -67864, 89552, -117632, 153926, -200352, 259904, -335872, 432336, -554952, 709728, -904784, 1150142, -1457136

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * (f(-q) * f(-q^16) / (f(q) * f(q^4)))^2 = q * (chi(-q^2) * chi(-q^4) / (chi(q) * chi(-q^8))^2)^2 in powers of q where chi(), f() are Ramanujan theta functions.

Expansion of (eta(q) * eta(q^4) * eta(q^16))^4 / (eta(q^2) * eta(q^8))^6 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)) = f(t) where q = exp(2 Pi i t).

a(n) = -(-1)^n * A215348(n). a(2*n) = -4 * A107035(n). Convolution inverse of A214035.

a(n) ~ -(-1)^n * exp(sqrt(n)*Pi) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

Example

q - 4*q^2 + 8*q^3 - 16*q^4 + 30*q^5 - 48*q^6 + 80*q^7 - 128*q^8 + 197*q^9 + ...

Mathematica

a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, -q]*EllipticTheta[2, 0, q^4]/(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^4]))/2, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Jan 07 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^16 + A))^4 / (eta(x^2 + A) * eta(x^8 + A))^6, n))}

Crossrefs

Cf. A107035, A214035, A215348.

Sequence in context: A099992 A301149 A301143 * A215348 A301144 A298803

Adjacent sequences: A215346 A215347 A215348 * A215350 A215351 A215352

Keyword

sign

Author

Michael Somos, Aug 08 2012

Status

approved