A261202 Expansion of phi(-x) * phi(-x^9) / f(-x^6)^2 in powers of x where phi(), f() are Ramanujan theta functions.

1, -2, 0, 0, 2, 0, 2, -4, 0, -4, 8, 0, 5, -14, 0, -8, 20, 0, 14, -28, 0, -20, 44, 0, 28, -66, 0, -40, 90, 0, 56, -124, 0, -80, 176, 0, 109, -244, 0, -144, 326, 0, 198, -432, 0, -268, 580, 0, 349, -772, 0, -456, 1004, 0, 600, -1300, 0, -780, 1692, 0, 1001

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 q^(1/2) * eta(q)^2 * eta(q^9)^2 / (eta(q^2) * eta(q^6)^2 * eta(q^18)) in powers of q.

Euler transform of period 18 sequence [ -2, -1, -2, -1, -2, 1, -2, -1, -4, -1, -2, 1, -2, -1, -2, -1, -2, 0, ...].

a(3*n + 2) = 0.

Example

G.f. = 1 - 2*x + 2*x^4 + 2*x^6 - 4*x^7 - 4*x^9 + 8*x^10 + 5*x^12 + ...
G.f. = 1/q - 2*q + 2*q^7 + 2*q^11 - 4*q^13 - 4*q^17 + 8*q^19 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^9] / QPochhammer[ x^6]^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^9 + A)^2 / (eta(x^2 + A) * eta(x^6 + A)^2 * eta(x^18 + A)), n))};

Crossrefs

Sequence in context: A046113 A262938 A143068 * A334596 A291900 A263146

Adjacent sequences: A261199 A261200 A261201 * A261203 A261204 A261205

Keyword

sign

Author

Michael Somos, Aug 11 2015

Status

approved