A214303 Expansion of f(-x^2, -x^4) * f(x^1, x^7) in powers of x where f(,) is Ramanujan's two-variable theta function.

1, 1, -1, -1, -1, -1, 0, 1, 0, -1, 2, 0, -1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, -1, -1, -1, 1, 0, -1, -1, -3, 1, 0, -1, 0, 1, 0, 0, 0, -1, 1, 0, 0, 1, 2, -1, -1, 0, -1, 0, 0, 1, 1, 0, 1, -1, -1, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, -1, -1, 0, 0, 0, 1, -1, 0, -1, -1

Offset: 0

Author

Michael Somos, Jul 11 2012

Keywords

• sign

Comments

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

a(n) = sum of (-1)^((u-1)/6) over all solutions of 48*n + 31 = 4*u^2 + 3*v^2 in integers where u == 1 (mod 6) and v == 3 (mod 8).

Examples

			1 + x - x^2 - x^3 - x^4 - x^5 + x^7 - x^9 + 2*x^10 - x^12 + x^15 + x^17 + ...
q^31 + q^79 - q^127 - q^175 - q^223 - q^271 + q^367 - q^463 + 2*q^511 + ...
		

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, A143379, A214263.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^8] QPochhammer[ -q^1, q^8] QPochhammer[ -q^7, q^8], {q, 0, n}]

PARI

{a(n) = local(s, v); if( n<0, 0, n = 48*n + 31; forstep( u=1, sqrtint( n\4), 2, if( u%3 && issquare( (n - 4*u^2)/3, &v), s += (-1)^((u+1)\6))); s)}

Formula

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

G.f.: (Sum_{k} (-1)^k * x^(3*k^2 + k)) * (Sum_{k} x^(4*k^2 + 3*k)).

a(n) = - A143379(2*n+1).