A139137 Expansion of phi(q) / phi(q^3) in powers of q where phi() is a Ramanujan theta function.

1, 2, 0, -2, -2, 0, 4, 4, 0, -6, -8, 0, 10, 12, 0, -16, -18, 0, 24, 28, 0, -36, -40, 0, 52, 58, 0, -74, -84, 0, 104, 116, 0, -144, -160, 0, 198, 220, 0, -268, -296, 0, 360, 396, 0, -480, -528, 0, 634, 694, 0, -832, -908, 0, 1084, 1184, 0, -1404, -1528, 0, 1808, 1964, 0, -2316, -2514, 0, 2952, 3196

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 f(q, -q^2) / f(-q, q^2) in powers of q where f(,) is Ramanujan's two-variable theta function. - Michael Somos, Apr 04 2015

Expansion of eta(q^2)^5 * eta(q^3)^2 * eta(q^12)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)^5) in powers of q.

Euler transform of period 12 sequence [ 2, -3, 0, -1, 2, 0, 2, -1, 0, -3, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132002. - Michael Somos, Apr 04 2015

G.f.: (Sum_{k in Z} x^k^2) / (Sum_{k in Z} x^(3*k^2)).

G.f.: Product_{k>0} P(12, x^k)^2 / (P(3, x^k) * P(6, x^k)^3) where P(n, x) is n-th cyclotomic polynomial.

Convolution inverse of A132002. - Michael Somos, Apr 04 2015

a(n) = (-1)^n * A252706(n). - Michael Somos, Apr 04 2015

a(3*n + 2) = 0. a(3*n) = A132002(n). a(3*n + 1) = 2 * A139135(n).

Example

G.f. = 1 + 2*q - 2*q^3 - 2*q^4 + 4*q^6 + 4*q^7 - 6*q^9 - 8*q^10 + 10*q^12 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3], {q, 0, n}]; (* Michael Somos, Apr 04 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5), n))};

Crossrefs

Cf. A132002, A139135, A252706.

Sequence in context: A138021 A166065 A252706 * A138231 A155100 A076880

Adjacent sequences: A139134 A139135 A139136 * A139138 A139139 A139140

Keyword

sign

Author

Michael Somos, Apr 10 2008

Status

approved