A139135 Expansion of psi(-q^3) / f(q) where psi(), f() are Ramanujan theta functions.

1, -1, 2, -4, 6, -9, 14, -20, 29, -42, 58, -80, 110, -148, 198, -264, 347, -454, 592, -764, 982, -1257, 1598, -2024, 2554, -3206, 4010, -5000, 6208, -7684, 9484, -11664, 14306, -17501, 21346, -25972, 31526, -38170, 46112, -55588, 66861, -80258, 96154, -114968, 137212

Offset

0,3

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (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^(-1/3) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2)^3 * eta(q^6)) in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A139136.

a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

Example

q - q^4 + 2*q^7 - 4*q^10 + 6*q^13 - 9*q^16 + 14*q^19 - 20*q^22 + 29*q^25 + ...

Mathematica

A139135[n_] := SeriesCoefficient[(QPochhammer[q]* QPochhammer[q^3]*QPochhammer[q^4]*QPochhammer[q^12])/(QPochhammer[q^2]^3 *QPochhammer[q^6]), {q, 0, n}]; Table[A139135[n], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)

Prog

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

Crossrefs

A139136(3*n + 1) = - a(n). A139137(3*n + 1) = 2 * a(n).

Apart from signs, same as A097197.

Sequence in context: A069916 A153140 A295341 * A097197 A260600 A119737

Adjacent sequences: A139132 A139133 A139134 * A139136 A139137 A139138

Keyword

sign

Author

Michael Somos, Apr 10 2008

Status

approved