A143434 Expansion of f(x, -x^3) in powers of x where f(,) is Ramanujan's two-variable 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

Examples

			1 + x - x^3 - x^6 - x^10 - x^15 + x^21 + x^28 + x^36 + x^45 - x^55 - x^66 + ...
q + q^9 - q^25 - q^49 - q^81 - q^121 + q^169 + q^225 + q^289 + q^361 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A143433.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), I x^2], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> I x}, {x, 0, n}]]

PARI

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

PARI

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

Formula

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

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

a(n) = (-1)^n * A143433(n).