A062157 a(n) = 0^n - (-1)^n.

0, 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, -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

0,1

Comments

Also the numerators of the series expansion of log(1+x). Denominators are A028310. - Robert G. Wilson v, Aug 14 2015

Links

Table of n, a(n) for n=0..101.

Wikipedia, Dirichlet eta function

Index entries for linear recurrences with constant coefficients, signature (-1).

Formula

a(n) = A000007(n) - A033999(n) = A062160(0, n).

G.f.: x/(1+x).

Euler transform of length 2 sequence [-1, 1]. - Michael Somos, Jul 05 2009

Moebius transform is length 2 sequence [1, -2]. - Michael Somos, Jul 05 2009

a(n) is multiplicative with a(2^e) = -1 if e > 0, a(p^e) = 1 if p > 2. - Michael Somos, Jul 05 2009

Dirichlet g.f.: zeta(s) * (1 - 2^(1-s)). - Michael Somos, Jul 05 2009

Also, Dirichlet g.f.: eta(s). - Ralf Stephan, Mar 25 2015

E.g.f.: 1 - exp(-x). - Alejandro J. Becerra Jr., Feb 16 2021

Example

G.f. = x - x^2 + x^3 - x^4 + x^5 - x^6 + x^7 - x^8 + x^9 - x^10 + ... - Michael Somos, Feb 20 2024

Mathematica

PadRight[{0}, 120, {-1, 1}] (* Harvey P. Dale, Aug 20 2012 *)
Join[{0}, LinearRecurrence[{-1}, {1}, 101]] (* Ray Chandler, Aug 12 2015 *)
f[n_] := 0^n - (-1)^n; f[0] = 0; Array[f, 105, 0] (* or *)
CoefficientList[ Series[ x/(1 + x), {x, 0, 80}], x] (* or *)
Numerator@ CoefficientList[ Series[ Log[1 + x], {x, 0, 80}], x] (* Robert G. Wilson v, Aug 14 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, -(-1)^n )}; /* Michael Somos, Jul 05 2009 */
(Magma) [0^n-(-1)^n: n in [0..100]] /* or */ [0] cat &cat[ [1, -1]: n in [1..80] ];; // Vincenzo Librandi, Aug 15 2015

Crossrefs

Convolution inverse of A019590.

Cf. A000007, A028310, A033999, A062160.

Sequence in context: A165596 A226523 A070238 * A103131 A112347 A057427

Adjacent sequences: A062154 A062155 A062156 * A062158 A062159 A062160

Keyword

easy,sign,mult

Author

Henry Bottomley, Jun 08 2001

Status

approved