A113681 Expansion of f(-x^2, -x^3)^2 / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset

0,1

Comments

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

f(a,b) = Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function.

|a(n)| is the characteristic function of A093722.

The exponents in the q-series for this sequence are the squares of the numbers of A057538.

This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^4, b = -x.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Quintuple Product Identity

Formula

Expansion of f(-x^7, -x^8) + x * f(-x^2, -x^13) where f() is Ramanujan's two-variable theta function.

Euler transform of period 5 sequence [ 1, -1, -1, 1, -1, ...].

G.f.: Sum_{k} (-1)^k * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)).

G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)) / ((1 - x^(5*k - 1)) * (1 - x^(5*k - 4))).

A010815(5*n) = a(n).

Example

1 + x - x^3 - x^7 - x^8 - x^14 + x^20 + x^29 + x^31 + x^42 - x^52 - x^66 - ...
q + q^121 - q^361 - q^841 - q^961 - q^1681 + q^2401 + q^3481 + q^3721 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^5] QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5])^2 / QPochhammer[ q], {q, 0, n}] (* Michael Somos, Jul 17 2012 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^((k%5==0) - kronecker( 5, k)), 1 + x * O(x^n)), n))}
(PARI) {a(n) = n*=5; if( issquare( 24*n + 1, &n), kronecker( 12, n))}

Crossrefs

Cf. A010815, A057538, A093722, A113430.

Sequence in context: A208546 A113430 A214529 * A356163 A295895 A179416

Adjacent sequences: A113678 A113679 A113680 * A113682 A113683 A113684

Keyword

sign

Author

Michael Somos, Nov 04 2005

Status

approved