A132319 Expansion of q^-1 * (chi(-q) * chi(-q^7))^3 in powers of q where chi() is a Ramanujan theta function.

1, -3, 3, -4, 9, -12, 15, -24, 39, -52, 66, -96, 130, -168, 219, -292, 390, -492, 625, -804, 1023, -1280, 1599, -2016, 2508, -3096, 3807, -4688, 5760, -7020, 8532, -10368, 12585, -15156, 18213, -21912, 26287, -31404, 37410, -44584, 53004, -62784, 74245, -87768

Offset

-1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (eta(q) * eta(q^7) / (eta(q^2) * eta(q^14)))^3 in powers of q.

Euler transform of period 14 sequence [ -3, 0, -3, 0, -3, 0, -6, 0, -3, 0, -3, 0, -3, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * v - v^2 + 8 * u + 6 * u * v.

G.f. is a period 1 Fourier series which satisfies f(-1 / (14 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A120006.

G.f.: x^-1 * (Product_{k>0} (1 + x^k) * (1 + x^(7*k)))^-3.

a(n) = A058503(n) unless n = 0. Convolution inverse is A120006.

a(n) = -(-1)^n * exp(2*Pi*sqrt(n/7)) / (2*7^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

G.f. = 1/q - 3 + 3*q - 4*q^2 + 9*q^3 - 12*q^4 + 15*q^5 - 24*q^6 + 39*q^7 - ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q, q^2] QPochhammer[ q^7, q^14])^3 / q, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)))^3, n))};

Crossrefs

Cf. A058503, A120006.

Sequence in context: A065678 A022598 A107635 * A130626 A175796 A115284

Adjacent sequences: A132316 A132317 A132318 * A132320 A132321 A132322

Keyword

sign

Author

Michael Somos, Aug 18 2007

Status

approved