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

Author

Michael Somos, Jul 02 2015

Keywords

• nonn

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).

Examples

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

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

Crossrefs

Cf. A122859, A132974, A132979, A258093.

Programs

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 *)

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))};

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