A113431 Expansion of f(-x) * f(-x^10) / f(-x^4, -x^6) in powers of x where f() is Ramanujan's two-variable theta function.

1, -1, -1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0, 0, -1, 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^3, b = x^2.

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

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^4, x^11) - x * f(x, x^14) = f(-x^5, -x^10) * f(-x, -x^4) / f(x^2, x^3) in powers of x where f() is Ramanujan's two-variable theta function.

Expansion of H(x^2) * f(-x) where H() is g.f. of A003106.

Euler transform of period 10 sequence [ -1, -1, -1, 0, -1, 0, -1, -1, -1, -1, ...].

G.f.: Sum_{k} x^(5*k * (3*k + 1) / 2) * (x^(-6*k) - x^(6*k + 2)).

G.f.: Product_{k>0} (1 - x^k) / ((1 - x^(10*k - 4)) * (1 - x^(10*k - 6))).

Example

1 - x - x^2 + x^4 + x^11 - x^15 - x^18 + x^23 + x^37 - x^44 - x^49 + ...
q^49 - q^169 - q^289 + q^529 + q^1369 - q^1849 - q^2209 + q^2809 + q^4489 + ...

Mathematica

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A113431[n_] := SeriesCoefficient[f[-x, -x^2]*f[-x^10, -x^20]/f[-x^4, -x^6], {x, 0, n}]; Table[A113431[n], {n, 0, 50}] (* G. C. Greubel, Jun 17 2017 *)

Prog

(PARI) {a(n) = local(m); if( n<0 || !issquare( n*120 + 49, &m) || 1! = gcd(m, 30),  0, (-1)^( m%30 \ 10))}
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 - x^k*[ 1, 1, 1, 1, 0, 1, 0, 1, 1, 1][k%10 + 1], 1 + x * O(x^n)), n))}

Crossrefs

Cf. A003106.

Sequence in context: A185906 A071003 A071002 * A116915 A266178 A339653

Adjacent sequences: A113428 A113429 A113430 * A113432 A113433 A113434

Keyword

sign

Author

Michael Somos, Oct 31 2005

Status

approved