A185125 Expansion of f(-x, x^5) in powers of x where f(,) is the Ramanujan general theta function.

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

Offset: 0

Author

Michael Somos, Jan 20 2012

Keywords

• sign

Comments

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

a(n) is nonzero if and only if n is a number of A001082.

The exponents in the q-series for this sequence are the squares of the numbers of A001651.

Examples

			G.f. = 1 - x + x^5 - x^8 - x^16 + x^21 - x^33 + x^40 + x^56 - x^65 + x^85 + ...
G.f. = q - q^4 + q^16 - q^25 - q^49 + q^64 - q^100 + q^121 + q^169 - q^196 + ...
		

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

Crossrefs

Cf. A010815, A080902, A185124.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x^6] QPochhammer[ -x^5, -x^6] QPochhammer[ -x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)

PARI

{a(n) = my(m); if( issquare( 3*n + 1, &m), (m%3!=0) * (-1)^((m+3) \ 6 + n), 0)};

Formula

Euler transform of period 24 sequence [ -1, 0, 0, 0, 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 0, 0, 0, -1, -1, ...].

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

a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(4*n + 1) = - A080902(n). a(8*n) = A010815(n).

a(n) = (-1)^n * A185124(n). - Michael Somos, Jun 30 2015