A260058 Expansion of f(x^2, x^4) * f(x^3, x^3) / f(-x, -x^2)^2 in power of x where f(, ) is Ramanujan's general theta function.

1, 2, 6, 14, 30, 60, 114, 208, 366, 626, 1044, 1704, 2730, 4300, 6672, 10212, 15438, 23076, 34134, 50008, 72612, 104560, 149400, 211920, 298554, 417902, 581412, 804254, 1106448, 1514316, 2062332, 2795488, 3772302, 5068632, 6782508, 9040224, 12004014, 15881692

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^4) * eta(q^6)^7 / (eta(q)^2 * eta(q^2) * eta(q^3)^2 * eta(q^12)^3) in powers of q.

Euler transform of period 12 sequence [ 2, 3, 4, 2, 2, -2, 2, 2, 4, 3, 2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187145.

a(n) = A261154(3*n).

a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Mar 17 2018

Example

G.f. = 1 + 2*x + 6*x^2 + 14*x^3 + 30*x^4 + 60*x^5 + 114*x^6 + 208*x^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^6] QPochhammer[ -x^4, x^6] QPochhammer[ x^6] EllipticTheta[ 3, 0, x^3] / QPochhammer[ x]^2, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x + A)^2 * eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^3), n))};

Crossrefs

Cf. A261154.

Sequence in context: A366542 A192966 A339668 * A331699 A327048 A228038

Adjacent sequences: A260055 A260056 A260057 * A260059 A260060 A260061

Keyword

nonn

Author

Michael Somos, Nov 08 2015

Status

approved