A214529 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, 0

Offset

0,1

Comments

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

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

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 - 1)/2) + floor(k/4)) * x^(3*k * (5*k + 1)/2) * (x^(4*k + 1) + x^(-16*k + 7)).

a(n) = (-1)^n * A208546(n).

Example

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

Mathematica

a[ n_] := Module[ {m}, If[ n >= 0 && OddQ[ DivisorSigma[ 0, 120 n + 1]], m = Sqrt[ 120 n + 1]; (-1)^(Quotient[ m, 40] + Quotient[ m, 3]), 0]]; Table[a[n], {n, 0, 30}]

Prog

(PARI) {a(n) = local(m); if( issquare( 120*n + 1, &m), (-1)^(m \ 40 + m \ 3))}

Crossrefs

Cf. A057538, A093722, A208546.

Sequence in context: A207735 A208546 A113430 * A113681 A356163 A295895

Adjacent sequences: A214526 A214527 A214528 * A214530 A214531 A214532

Keyword

sign

Author

Michael Somos, Jul 20 2012

Status

approved