A193779 Expansion of f(-x) * f(-x^15) / (f(-x^3) * f(-x^5)) in powers of x where f() is a Ramanujan theta function.

1, -1, -1, 1, -1, 1, 1, -2, 0, 2, 0, -1, 1, -2, -1, 5, -3, -2, 5, -4, 1, 4, -7, 0, 6, -3, -3, 6, -6, -2, 15, -12, -6, 15, -12, 3, 15, -20, -2, 20, -11, -7, 19, -20, -7, 40, -29, -14, 40, -34, 3, 40, -48, -5, 52, -33, -17, 52, -50, -14, 93, -74, -32, 97, -80, 3, 99, -112, -15

Offset

0,8

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 q^(-1/3) * eta(q) * eta(q^15) / (eta(q^3) * eta(q^5)) in powers of q.

Euler transform of period 15 sequence [ -1, -1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, ...].

Given g.f. A(x), then B(x) = x * A(x^3) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (v - u^2) * (u - v^2) - 2 * u^2 * v^2.

A058685(3*n + 1) = - a(n). Convolution inverse of A058686.

Example

1 - x - x^2 + x^3 - x^4 + x^5 + x^6 - 2*x^7 + 2*x^9 - x^11 + x^12 + ...
q - q^4 - q^7 + q^10 - q^13 + q^16 + q^19 - 2*q^22 + 2*q^28 - q^34 + q^37 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/3)* eta[q]*eta[q^15]/(eta[q^3]*eta[q^5]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 03 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^15 + A) / (eta(x^3 + A) * eta(x^5 + A)), n))}

Crossrefs

Cf. A058685, A058686.

Sequence in context: A287341 A282432 A046922 * A279048 A263485 A263489

Adjacent sequences: A193776 A193777 A193778 * A193780 A193781 A193782

Keyword

sign

Author

Michael Somos, Aug 04 2011

Status

approved