A298733 Expansion of phi(-x^9) * f(-x^3)^2 / f(-x^2)^3 in powers of x where f(), phi() are Ramanujan theta functions.

1, 0, 3, -2, 9, -6, 21, -18, 48, -44, 99, -102, 204, -216, 393, -438, 747, -846, 1362, -1584, 2448, -2872, 4275, -5082, 7356, -8784, 12390, -14894, 20592, -24798, 33651, -40644, 54336, -65640, 86535, -104628, 136356, -164736, 212388, -256498, 327690, -395214

Offset

0,3

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^3)^2 * eta(q^9)^2 / (eta(q^2)^3 * eta(q^18)) in powers of q.

Euler transform of period 18 sequence [0, 3, -2, 3, 0, 1, 0, 3, -4, 3, 0, 1, 0, 3, -2, 3, 0, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 4/9 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A182036.

a(n) ~ (-1)^n * exp(2^(3/2)*Pi*sqrt(n)/3) / (2^(3/4) * 3^(5/2) * n^(3/4)). - Vaclav Kotesovec, Mar 21 2018

Example

G.f. = 1 + 3*x^2 - 2*x^3 + 9*x^4 - 6*x^5 + 21*x^6 - 18*x^7 + 48*x^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^9] QPochhammer[ x^3]^2 / QPochhammer[ x^2]^3, {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^9 + A)^2 / (eta(x^2 + A)^3 * eta(x^18 + A)), n))};
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)^2*eta(q^9)^2/(eta(q^2)^3*eta(q^18)))} \\ Altug Alkan, Mar 21 2018

Crossrefs

Cf. A182036.

Sequence in context: A270199 A349579 A082819 * A197311 A078478 A346109

Adjacent sequences: A298730 A298731 A298732 * A298734 A298735 A298736

Keyword

sign

Author

Michael Somos, Jan 29 2018

Status

approved