A263041 Expansion of f(-x, -x^5)^2 / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

1, -3, 4, -5, 8, -14, 20, -25, 37, -54, 71, -91, 121, -164, 210, -264, 343, -443, 554, -687, 863, -1087, 1340, -1637, 2021, -2489, 3027, -3659, 4442, -5391, 6480, -7755, 9306, -11153, 13278, -15752, 18711, -22203, 26214, -30860, 36354, -42777, 50137, -58628

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(-x)^3 * psi(x^3)^2 / f(-x^2)^4 in powers of x where psi(), f() are Ramanujan theta functions.

Expansion of q^(-13/24) * eta(q)^3 * eta(q^6)^4 / (eta(q^2)^4 * eta(q^3)^2) in powers of q.

Euler transform of period 6 sequence [ -3, 1, -1, 1, -3, -1, ...].

a(n) = - A053269(3*n + 2).

a(n) ~ (-1)^n * exp(sqrt(n/2)*Pi) / (6*sqrt(n)). - Vaclav Kotesovec, Apr 17 2016

Example

G.f. = 1 - 3*x + 4*x^2 - 5*x^3 + 8*x^4 - 14*x^5 + 20*x^6 - 25*x^7 + 37*x^8 + ...
G.f. = q^13 - 3*q^37 + 4*q^61 - 5*q^85 + 8*q^109 - 14*q^133 + 20*q^157 - 25*q^181 + ...

Mathematica

a[ n_] := SeriesCoefficient[ x^(-1/2) QPochhammer[ x] (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)^3 * eta(x^6 + A)^4 / (eta(x^2 + A)^4 * eta(x^3 + A)^2), n))};

Crossrefs

Cf. A053269.

Sequence in context: A368800 A039020 A055742 * A216888 A362218 A106048

Adjacent sequences: A263038 A263039 A263040 * A263042 A263043 A263044

Keyword

sign

Author

Michael Somos, Apr 17 2016

Status

approved