A215597 Expansion of psi(-x) * f(-x)^3 in powers of x where psi(), f() are Ramanujan theta functions.

1, -4, 3, 4, -2, 0, -11, 4, 0, 12, 10, -12, -7, -4, 0, -12, 16, 0, 6, 0, 9, 8, -10, 0, -18, -20, 0, 20, -14, 12, 11, 24, 0, 0, -22, 0, 16, -20, -6, -12, 0, 0, -3, 4, 0, -20, 48, 0, 14, 28, 0, -40, 0, 0, 0, -8, -33, -4, -26, 0, 30, 28, 0, 0, 2, 12, -16, 20, 0

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^(-1/4) * eta(q)^4 * eta(q^4) / eta(q^2) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(19/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215596.

a(n) = (-1)^floor( n/2 ) * b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-p)^(e/2) if p == 3 (mod 4),

Convolution of A106459 and A010816.

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * exp(Pi / 4) * Pi * 2^(1/8) / Gamma(3/4)^4 = A388770. - Simon Plouffe, Sep 18 2025

Example

1 - 4*x + 3*x^2 + 4*x^3 - 2*x^4 - 11*x^6 + 4*x^7 + 12*x^9 + 10*x^10 + ...
q - 4*q^5 + 3*q^9 + 4*q^13 - 2*q^17 - 11*q^25 + 4*q^29 + 12*q^37 + 10*q^41 + ...

Mathematica

A215597[n_] := SeriesCoefficient[(QPochhammer[x]^4 * QPochhammer[x^4])/ QPochhammer[x^2], {x, 0, n}]; Table[A215597[n], {n, 0, 50}] (* G. C. Greubel, Oct 01 2017 *)

Prog

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

Crossrefs

Cf. A010816, A106459, A215596.

Sequence in context: A170987 A239735 A196826 * A266110 A204819 A161882

Adjacent sequences: A215594 A215595 A215596 * A215598 A215599 A215600

Keyword

sign

Author

Michael Somos, Aug 16 2012

Status

approved