A206959 Expansion of f(-x^5, x^7) + x * f(x, -x^11) in powers of x where f() is Ramanujan's two-variable theta function.

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

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^8) * f(-x^8,-x^8) / f(-x,x^3) in powers of x where f() is Ramanujan't two-variable theta function.

Euler transform of period 16 sequence [ 1, 0, -1, 0, -1, 1, 1, -2, 1, 1, -1, 0, -1, 0, 1, -1, ...].

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

G.f.: Product_{k>0} (1 + (-1)^k * x^(4*k-1)) * (1 - (-1)^k * x^(4*k-3)) * (1 - (-1)^k * x^(4*k)) * (1 + x^(8*k-6)) * (1 + x^(8*k-2)).

a(5*n + 3) = a(5*n + 4) = 0. |a(n)| = A080995(n).

a(n) = (-1)^n * A206958(n). - Michael Somos, Apr 01 2015

Example

G.f. = 1 + x + x^2 - x^5 + x^7 - x^12 - x^15 - x^22 - x^26 - x^35 - x^40 + x^51 - ...
G.f. = q + q^25 + q^49 - q^121 + q^169 - q^289 - q^361 - q^529 - q^625 - q^841 - ...

Mathematica

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 0, 1, 0, 1, -1, -1, 2, -1, -1, 1, 0, 1, 0, -1, 1}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Apr 01 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^12] (QPochhammer[ x^5, -x^12] QPochhammer[ -x^7, -x^12] + x QPochhammer[ -x, -x^12] QPochhammer[ x^11, -x^12]), {x, 0, n}]; (* Michael Somos, Apr 01 2015 *)

Prog

(PARI) {a(n) = my(m); if( issquare( 24*n + 1, &m), if( m%6 != 5, m = -m); m \= 6; (-1)^((-m) \ 4), 0)};

Crossrefs

Cf. A080995, A206958.

Sequence in context: A143062 A010815 A206958 * A080995 A121373 A199918

Adjacent sequences: A206956 A206957 A206958 * A206960 A206961 A206962

Keyword

sign

Author

Michael Somos, Feb 14 2012

Status

approved