A260574 Expansion of phi(x^3) * psi(x^3) / f(x) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

1, -1, 2, 0, 2, -1, 4, -2, 5, -2, 6, -2, 10, -4, 12, -4, 15, -6, 20, -8, 26, -9, 32, -12, 40, -16, 50, -18, 60, -22, 76, -28, 92, -33, 110, -40, 134, -50, 160, -58, 191, -70, 230, -84, 272, -98, 320, -116, 380, -138, 446, -160, 522, -188, 612, -222, 715, -256

Offset

0,3

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^3)^3 / (f(x) * phi(-x^6)) in powers of x where phi(), f() are Ramanujan theta functions.

Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^3 * eta(q^12)^2) in powers of q.

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

a(2*n) = A085140(n). a(2*n + 1) = - A097196(n). a(4*n + 1) = - A257655(n).

Example

G.f. = 1 - x + 2*x^2 + 2*x^4 - x^5 + 4*x^6 - 2*x^7 + 5*x^8 - 2*x^9 + ...
G.f. = q - q^4 + 2*q^7 + 2*q^13 - q^16 + 4*q^19 - 2*q^22 + 5*q^25 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3]^3 / (QPochhammer[ -x] EllipticTheta[ 4, 0, x^6]), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^3 * eta(x^12 + A)^2), n))};
(PARI) q='q+O('q^99); Vec(eta(q)*eta(q^4)*eta(q^6)^7/(eta(q^2)^3*eta(q^3)^3*eta(q^12)^2)) \\ Altug Alkan, Aug 01 2018

Crossrefs

Cf. A085140, A097196, A257655.

Sequence in context: A353508 A029181 A261426 * A058707 A242691 A081082

Adjacent sequences: A260571 A260572 A260573 * A260575 A260576 A260577

Keyword

sign

Author

Michael Somos, Jul 29 2015

Status

approved