A248886 Expansion of f(-x, -x) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

1, -2, 1, -2, 3, -2, 2, 0, 2, -2, 1, -4, 0, -2, 3, -2, 2, 0, 4, -2, 2, 0, 0, -2, 1, -4, 2, -2, 2, -2, 3, -2, 0, -2, 2, -2, 2, 0, 2, -4, 4, 0, 0, 0, 1, -2, 4, 0, 2, -4, 2, -2, 1, -6, 0, -2, 2, 0, 0, -2, 4, -2, 0, -2, 2, 0, 4, 0, 4, -2, 1, -2, 0, -2, 4, 0, 0, -2

Offset

0,2

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 = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of f(-x)^2 * phi(-x^6) / phi(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.

Expansion of phi(-x) * phi(-x^6) / chi(-x^2) in powers of q where phi(), chi() are Ramanujan theta functions.

Expansion of q^(-1/12) * eta(q)^2 * eta(q^4) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [-2, 0, -2, -1, -2, -2, -2, -1, -2, 0, -2, -2, ...].

a(n) = (-1)^n * A123884(n). a(2*n) = A131961(n). a(2*n + 1) = -2 * A131963(n).

Example

G.f. = 1 - 2*x + x^2 - 2*x^3 + 3*x^4 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + ...
G.f. = q - 2*q^13 + q^25 - 2*q^37 + 3*q^49 - 2*q^61 + 2*q^73 + 2*q^97 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^2], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2], {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A)^2* eta(x^12 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q)^2*eta(q^4)*eta(q^6)^2/(eta(q^2)^2*eta(q^12))) \\ Altug Alkan, Jul 31 2018

Crossrefs

Cf. A123884, A131961, A131963.

Sequence in context: A141662 A328383 A088062 * A123884 A178412 A182598

Adjacent sequences: A248883 A248884 A248885 * A248887 A248888 A248889

Keyword

sign

Author

Michael Somos, Oct 01 2015

Status

approved