A207710 Expansion of f(x) * f(-x^10) / f(-x^2, -x^8) 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

Offset

0,1

Comments

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

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Quintuple Product Identity

Formula

Expansion of f(x^5, -x^10) * f(-x^2, x^3) / f(-x, x^4) = f(-x^7, x^8) + x * f(x^2, -x^13) in powers of x where f() is Ramanujan's two-variable theta function.

Euler transform of period 20 sequence [ 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 1, -1, ...].

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

The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.

G.f.: Prod_{k>0} (1 - (-x)^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))).

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

a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = 0. a(n) * (-1)^n = A113430(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

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A207710[n_] := SeriesCoefficient[f[x^5, -x^10]*f[-x^2, x^3]/f[-x, x^4], {x, 0, n}]; Table[A207710[n], {n, 0, 50}] (* G. C. Greubel, Jun 18 2017 *)

Prog

(PARI) {a(n) = local(m); if( issquare( 120*n + 1, &m), kronecker( -120, m) * (-1)^(m \ 15))}

Crossrefs

Cf. A057538, A093722, A113430.

Sequence in context: A053866 A143259 A359818 * A207735 A208546 A113430

Adjacent sequences: A207707 A207708 A207709 * A207711 A207712 A207713

Keyword

sign

Author

Michael Somos, Feb 19 2012

Status

approved