A260309 Expansion of phi(q) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, 0, 2, 0, -2, -4, 0, 2, -4, 0, 0, 0, 0, -4, 2, 0, 0, 0, 0, 0, -4, 0, 2, 6, 0, 0, 4, 0, 0, -4, 0, 4, 0, 0, 2, 0, 0, 0, 4, 0, -4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, -2, -8, 0, 0, -4, 0, 4, 0, 0, -4, 2, 0, 0, 0, 0, 0, -8, 0, 0, 4, 0, 0, 0, 0, 0, -4, 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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^2)^5 * eta(q^6)^2 / (eta(q)^2 * eta(q^4)^2* eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -5, 2, -1, 2, -3, 2, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 1536^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260308.

G.f.: Sum_{i,j in Z} (-1)^j * x^(i^2 + 6*j^2).

a(n) = (-1)^floor(n/2) * A046113(n). a(3*n + 2) = 0. a(4*n) = A046113(n). 2 * a(n) = A260110(n).

Example

G.f. = 1 + 2*x + 2*x^4 - 2*x^6 - 4*x^7 + 2*x^9 - 4*x^10 - 4*x^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, -q^6], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2* eta(x^12 + A)), n))};

Crossrefs

Cf. A046113, A260110, A260308.

Sequence in context: A213024 A291289 A095759 * A046113 A262938 A143068

Adjacent sequences: A260306 A260307 A260308 * A260310 A260311 A260312

Keyword

sign

Author

Michael Somos, Jul 22 2015

Status

approved