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a(n) = 0^n - (-1)^n.
17

%I #40 Feb 22 2024 09:04:48

%S 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,

%T -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,

%U 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

%N a(n) = 0^n - (-1)^n.

%C Also the numerators of the series expansion of log(1+x). Denominators are A028310. - _Robert G. Wilson v_, Aug 14 2015

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (-1).

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

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

%F Euler transform of length 2 sequence [-1, 1]. - _Michael Somos_, Jul 05 2009

%F Moebius transform is length 2 sequence [1, -2]. - _Michael Somos_, Jul 05 2009

%F a(n) is multiplicative with a(2^e) = -1 if e > 0, a(p^e) = 1 if p > 2. - _Michael Somos_, Jul 05 2009

%F Dirichlet g.f.: zeta(s) * (1 - 2^(1-s)). - _Michael Somos_, Jul 05 2009

%F Also, Dirichlet g.f.: eta(s). - _Ralf Stephan_, Mar 25 2015

%F E.g.f.: 1 - exp(-x). - _Alejandro J. Becerra Jr._, Feb 16 2021

%e 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

%t PadRight[{0},120,{-1,1}] (* _Harvey P. Dale_, Aug 20 2012 *)

%t Join[{0},LinearRecurrence[{-1},{1},101]] (* _Ray Chandler_, Aug 12 2015 *)

%t f[n_] := 0^n - (-1)^n; f[0] = 0; Array[f, 105, 0] (* or *)

%t CoefficientList[ Series[ x/(1 + x), {x, 0, 80}], x] (* or *)

%t Numerator@ CoefficientList[ Series[ Log[1 + x], {x, 0, 80}], x] (* _Robert G. Wilson v_, Aug 14 2015 *)

%o (PARI) {a(n) = if( n<1, 0, -(-1)^n )}; /* _Michael Somos_, Jul 05 2009 */

%o (Magma) [0^n-(-1)^n: n in [0..100]] /* or */ [0] cat &cat[ [1, -1]: n in [1..80] ];; // _Vincenzo Librandi_, Aug 15 2015

%Y Convolution inverse of A019590.

%Y Cf. A000007, A028310, A033999, A062160.

%K easy,sign,mult

%O 0,1

%A _Henry Bottomley_, Jun 08 2001