A132002 Expansion of phi(q^3) / phi(q) in powers of q where phi() is a Ramanujan theta function.

1, -2, 4, -6, 10, -16, 24, -36, 52, -74, 104, -144, 198, -268, 360, -480, 634, -832, 1084, -1404, 1808, -2316, 2952, -3744, 4728, -5946, 7448, -9294, 11556, -14320, 17688, -21780, 26740, -32736, 39968, -48672, 59122, -71644, 86616, -104484, 125768, -151072

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

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v+u) * (v-u) + (1 - u*v) * (1 - 3*u*v).

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3*u*v^2 * (1 - u*v).

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 A139007. - Michael Somos, Apr 04 2015

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

G.f.: Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)). - Michael Somos, Apr 04 2015

a(n) = (-1)^n * A098151(n).

Convolution inverse of A139137. Convolution square is A261320. - Michael Somos, Aug 14 2015

Expansion of f(-q, q^2) / f(q, -q^2) in powers of q where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 02 2015

a(n) = A139136(3*n) = A139137(3*n). - Michael Somos, Nov 02 2015

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

G.f. = 1 - 2*x + 4*x^2 - 6*x^3 + 10*x^4 - 16*x^5 + 24*x^6 - 36*x^7 + 52*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Apr 04 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q^3] QPochhammer[ -q^2, -q^3] / (QPochhammer[ -q, -q^3] QPochhammer[ q^2, -q^3]), {q, 0, n}]; (* Michael Somos, Nov 02 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1 + x * O(x^n)) / sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n)), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

Crossrefs

Cf. A098151, A139007, A139136, A139137, A261320.

Sequence in context: A241903 A261204 A293422 * A098151 A137414 A211971

Adjacent sequences: A131999 A132000 A132001 * A132003 A132004 A132005

Keyword

sign

Author

Michael Somos, Aug 06 2007

Status

approved