A262771 Expansion of f(-x, -x^5) / f(x^2, x^4) in powers of x where f( , ) is Ramanujan's general theta function.

1, -1, -1, 1, 0, -1, 1, 0, 0, 1, -2, 0, 3, -2, -2, 4, 0, -3, 2, -1, -1, 3, -4, -1, 8, -5, -6, 10, -2, -7, 7, -2, -2, 8, -10, -3, 18, -12, -12, 22, -4, -15, 15, -5, -6, 19, -20, -7, 38, -24, -26, 45, -10, -30, 34, -13, -13, 40, -40, -16, 74, -46, -48, 87, -22

Offset

0,11

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..1500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/4) * eta(q) * eta(q^12) / (eta(q^3) * eta(q^4)) in powers of q.

Euler transform of period 12 sequence [-1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = f(t) where q = exp(2 Pi i t).

Convolution inverse of A112192.

Example

G.f. = 1 - x - x^2 + x^3 - x^5 + x^6 + x^9 - 2*x^10 + 3*x^12 + ...
G.f. = q - q^3 - q^5 + q^7 - q^11 + q^13 + q^19 - 2*q^21 + 3*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^12] / (QPochhammer[ x^3] QPochhammer[ x^4]), {x, 0, n}];
eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/4)* eta[q]*eta[q^12]/(eta[q^3]*eta[q^4]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 30 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A) / (eta(x^3 + A) * eta(x^4 + A)), n))};

Crossrefs

Cf. A112192.

Sequence in context: A368818 A110493 A118234 * A152039 A180196 A317843

Adjacent sequences: A262768 A262769 A262770 * A262772 A262773 A262774

Keyword

sign

Author

Michael Somos, Sep 30 2015

Status

approved