A210245 Signs of the polylogarithm li(-n,-1/2).

1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1

Offset

0

Comments

Given an integer n, consider the infinite series s(n) = li(-n,-1/2) = Sum_{k>=0} (-1)^k*k^n/2^k. Then the value of a(n) is the sign of s(n).

Should s(n) be 0, the sign would be set to 0 as well. However, it is not known whether this ever happens.

Conjecture: a(n) is the same as the sign of cos((n+1)*arctan(Pi/log(2))). - Mikhail Kurkov, Apr 13 2021 [Holds for all n<=10000. - Vaclav Kotesovec, Mar 02 2026]

Links

Stanislav Sykora, Table of n, a(n) for n = 0..199

Stanislav Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), Stan's Library Vol. I, April 2006, updated March 2012. See Eq.(29).

Eric Weisstein's World of Mathematics, Polylogarithm

Formula

a(n) = sign(A210244(n)) = sign(A212846(n)).

Example

a(5) = sign(A212846(5)) = sign(-7) = -1.

Mathematica

Join[{1}, Sign[PolyLog[-Range[80], -1/2]]] (* Harvey P. Dale, Jul 21 2024 *)

Crossrefs

Cf. A210244, A212846.

Sequence in context: A242179 A319117 A063747 * A210247 A256175 A319116

Adjacent sequences: A210242 A210243 A210244 * A210246 A210247 A210248

Keyword

sign

Author

Stanislav Sykora, Mar 19 2012

Status

approved