A208546 Expansion of f(-x^29, x^31) + x * f(-x^19, x^41) - x^3 * f(-x^11, x^49) + x^7 * f(x, -x^59) in powers of x where f() is Ramanujan's two-variable theta function.

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

Offset: 0

Author

Michael Somos, Feb 28 2012

Keywords

• sign

Examples

			G.f. = 1 + x - x^3 + x^7 + x^8 + x^14 - x^20 - x^29 + x^31 + x^42 - x^52 - x^66 + ...
G.f. = q + q^121 - q^361 + q^841 + q^961 + q^1681 - q^2401 - q^3481 + q^3721 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A057538, A093722.

Programs

Mathematica

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; CoefficientList[Series[f[-x^29, x^31] + x*f[-x^19, x^41] - x^3*f[-x^11, x^49] + x^7*f[x, -x^59], {x, 0, 50}], x] (* G. C. Greubel, Aug 11 2018 *)

PARI

{a(n) = local(m); if( issquare( 120*n + 1, &m), (-1)^(m \ 40 + m \ 12))}

Formula

|a(n)| is the characteristic function of A093722.

The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.

Euler transform of a period 80 sequence.

G.f.: Sum_{k} (-1)^floor(k/4) * x^(3*k * (5*k + 1)/2) * (x^(4*k + 1) + x^(-16*k + 7)).