A263050 Expansion of f(-x) * f(x^4, x^8) / f(-x^3)^2 in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, -1, 2, -1, -2, 4, -2, -4, 8, -4, -7, 14, -6, -13, 24, -10, -21, 40, -17, -35, 63, -26, -55, 98, -40, -84, 150, -61, -127, 224, -90, -189, 330, -131, -275, 480, -190, -397, 687, -270, -565, 974, -381, -795, 1367, -533, -1109, 1898, -737, -1533, 2614

Offset

0,4

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 q^(1/24) * eta(q) * eta(q^8) * eta(q^12)^2 / (eta(q^3)^2 * eta(q^4) * eta(q^24)) in powers of q.

Euler transform of period 24 sequence [-1, -1, 1, 0, -1, 1, -1, -1, 1, -1, -1, 0, -1, -1, 1, -1, -1, 1, -1, 0, 1, -1, -1, 0, ...].

a(n) = A137569(2*n).

Example

G.f. = 1 - x - x^2 + 2*x^3 - x^4 - 2*x^5 + 4*x^6 - 2*x^7 - 4*x^8 + 8*x^9 + ...
G.f. = 1/q - q^23 - q^47 + 2*q^71 - q^95 - 2*q^119 + 4*q^143 - 2*q^167 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x^4, x^4] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x^3]^2, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A)), n))};
(PARI) q='q+O('q^99); Vec(eta(q)*eta(q^8)*eta(q^12)^2/(eta(q^3)^2*eta(q^4)*eta(q^24))) \\ Altug Alkan, Jul 31 2018

Crossrefs

Cf. A137569.

Sequence in context: A120253 A307018 A274624 * A060547 A079878 A324469

Adjacent sequences: A263047 A263048 A263049 * A263051 A263052 A263053

Keyword

sign

Author

Michael Somos, Oct 08 2015

Status

approved