A143433 Expansion of f(-x, x^3) in powers of x where f(,) is Ramanujan's general theta function.

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

Offset: 0

Author

Michael Somos, Aug 14 2008

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x + x^3 - x^6 - x^10 + x^15 - x^21 + x^28 + x^36 - x^45 + x^55 + ...
G.f. = q - q^9 + q^25 - q^49 - q^81 + q^121 - q^169 + q^225 + q^289 + ...
		

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. A143434, A244465, A244525.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x^4] QPochhammer[ -x^3, -x^4] QPochhammer[ -x^4], {x, 0, n}]; (* Michael Somos, Jun 03 2015 *)

PARI

{a(n) = if( n<0, 0, if( issquare(8*n + 1, &n), n = n\2; (-1)^(n + n\4), 0))};

PARI

{a(n) = my(A); if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^( [1, 1, 0, -1, -1, -1, 1, 1, 2, 1, 1, -1, -1, -1, 0, 1, 1] [k%16 + 1]), 1 + x * O(x^n)), n))};

Formula

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

Pattern of signs of nonzero terms is A143431.

G.f.: Sum_{k>=0} (-1)^(k + floor(k/4)) * x^(k * (k+1) / 2).

a(n) = (-1)^n * A143434(n).

a(2*n) = A244465(n). a(2*n + 1) = - A244525(n). a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = 0.