A259662 Expansion of phi(-q^3) / phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

1, 6, 24, 78, 222, 576, 1392, 3180, 6936, 14550, 29520, 58176, 111750, 209820, 385968, 696960, 1237470, 2163456, 3728904, 6343068, 10658880, 17708412, 29108880, 47373696, 76378992, 122058870, 193435248, 304134558, 474609180, 735374016, 1131698448, 1730375436

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of 1 / (2*a(q^2) - a(q)) = b(q^2) / b(q)^2 in powers of q where a(), b() are cubic AGM theta functions.

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

Euler transform of period 6 sequence [ 6, 3, 4, 3, 6, 2, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w^2*(u + v)^2 - 2*u*v^2*(v+w).

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 432^(-1/2) (t/I)^-1 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A258093.

G.f.: Product_{k>0} (1 + x^k)^3 * (1 - x^(3*k)) / ((1 + x^(3*k)) * (1 - x^k)^3).

a(n) = A132974(2*n) = A132979(2*n).

Convolution inverse of A122859.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(5/4) * n^(5/4)). - Vaclav Kotesovec, Oct 14 2015

Example

G.f. = 1 + 6*x + 24*x^2 + 78*x^3 + 222*x^4 + 576*x^5 + 1392*x^6 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / EllipticTheta[ 4, 0, x]^3, {x, 0, n}];
nmax=60; CoefficientList[Series[Product[(1+x^k)^3 * (1-x^(3*k)) / ((1+x^(3*k)) * (1-x^k)^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)

Prog

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

Crossrefs

Cf. A122859, A132974, A132979, A258093.

Sequence in context: A180437 A276179 A162583 * A058809 A140088 A359133

Adjacent sequences: A259659 A259660 A259661 * A259663 A259664 A259665

Keyword

nonn

Author

Michael Somos, Jul 02 2015

Status

approved