A187150 Expansion of psi(-x)^4 / chi(-x)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

1, -2, 1, -2, 0, 4, 1, 2, -5, 0, -5, 4, 1, -2, -5, 0, 7, 4, 7, 0, -4, -10, 7, -8, 0, 4, 0, -8, 2, 0, 1, -2, 0, 2, 0, 14, 7, 0, -5, 10, -11, -8, -10, -2, 0, 10, -4, 4, 0, 0, -5, -8, -11, 10, 0, 0, 14, -2, 20, 0, -11, 4, 13, 2, -5, -14, 0, -14, 13, 0, -11, -14, 8, -2, 0, 10, 13, -18, 0, 0, -5

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 288 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187149.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/16) * exp(7*Pi/12) * Pi * 2^(3/4) / Gamma(3/4)^4 = A388705. - Simon Plouffe, Sep 18 2025

Example

G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^5 + x^6 + 2*x^7 - 5*x^8 - 5*x^10 + ...
G.f. = q^7 - 2*q^19 + q^31 - 2*q^43 + 4*q^67 + q^79 + 2*q^91 - 5*q^103 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^4]^2 / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, Pi/4, x^(1/2)]^4 / QPochhammer[ x, x^2]^2, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

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

Crossrefs

Cf. A187149.

Sequence in context: A395057 A229817 A080966 * A023895 A070963 A174064

Adjacent sequences: A187147 A187148 A187149 * A187151 A187152 A187153

Keyword

sign

Author

Michael Somos, Mar 05 2011

Status

approved