A217786 Expansion of (psi(x^3) / psi(x))^2 in powers of x where psi() is a Ramanujan theta function.

1, -2, 3, -4, 7, -12, 17, -24, 36, -52, 71, -96, 133, -182, 240, -316, 420, -552, 713, -916, 1182, -1516, 1920, -2424, 3063, -3852, 4806, -5976, 7430, -9204, 11336, -13924, 17088, -20908, 25473, -30960, 37586, -45518, 54939, -66172, 79603, -95556, 114399

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 q^(-1/2) * eta(q)^2 * eta(q^6)^4 / (eta(q^2)^4 * eta(q^3)^2) in powers of q.

Euler transform of period 6 sequence [ -2, 2, 0, 2, -2, 0, ...].

Given g.f. A(x), B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 - v) * (v - u^2) - 4 * u^2 * v.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A217771.

G.f.: Product_{k>0} (1 + x^k + x^(2*k))^2 * (1 - x^k + x^(2*k))^4.

Convolution square of A101195. Convolution inverse of A058487.

a(n) = - A139216(6*n + 3). - Michael Somos, Sep 07 2015

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (6^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Example

G.f. = 1 - 2*x + 3*x^2 - 4*x^3 + 7*x^4 - 12*x^5 + 17*x^6 - 24*x^7 + 36*x^8 + ...
G.f. = q - 2*q^3 + 3*q^5 - 4*q^7 + 7*q^9 - 12*q^11 + 17*q^13 - 24*q^15 + 36*q^17 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(-1/2) (EllipticTheta[ 2, 0, x^(3/2)] / EllipticTheta[ 2, 0, x^(1/2)])^2, {x, 0, n}];

Prog

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

Crossrefs

Cf. A058487, A101195, A139216, A217771.

Sequence in context: A249578 A325244 A376987 * A228494 A292324 A289919

Adjacent sequences: A217783 A217784 A217785 * A217787 A217788 A217789

Keyword

sign

Author

Michael Somos, Mar 24 2013

Status

approved