A181983 - OEIS

%I #78 Oct 24 2023 07:27:40

%S 0,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15,-16,17,-18,19,-20,21,

%T -22,23,-24,25,-26,27,-28,29,-30,31,-32,33,-34,35,-36,37,-38,39,-40,

%U 41,-42,43,-44,45,-46,47,-48,49,-50,51,-52,53,-54,55,-56,57,-58,59

%N a(n) = (-1)^(n+1) * n.

%C This is the Lucas U(-2,1) sequence. - _R. J. Mathar_, Jan 08 2013

%C Apparently the Mobius transform of A002129. - _R. J. Mathar_, Jan 08 2013

%C For n>0, a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = max(i,j) for 1 <= i,j <= n. - _Enrique Pérez Herrero_, Jan 14 2013

%C The sums of the terms of this sequence is the divergent series 1 - 2 + 3 - 4 + ... . Euler summed it to 1/4 which was one of the first examples of summing divergent series. - _Michael Somos_, Jun 05 2013

%H Enrique Pérez Herrero, <a href="http://psychedelic-geometry.blogspot.com/2013/01/max-determinant.html">Max Determinant</a>, 2013.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7">1 - 2 + 3 - 4 + ...</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-1).

%H <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>.

%F G.f.: x / (1 + x)^2.

%F E.g.f.: x / exp(x).

%F a(n) = -a(-n) = -(-1)^n * A001477(n) for all n in Z.

%F a(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = Bernoulli(k) for k = 0, 1, ..., n.

%F A001787(n) = p(0) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 1, ..., n+1. - _Michael Somos_, Jun 05 2013

%F Euler transform of length 2 sequence [-2, 2].

%F Series reversion of g.f. is A000108(n) (Catalan numbers) with a(0)=0.

%F Series reversion of e.g.f. is A000169. INVERT transform omitting a(0)=0 is A049347. PSUM transform is A001057. BINOMIAL transform is A154955. - _Michael Somos_, Jun 05 2013

%F n * a(n) = A162395(n). - _Michael Somos_, Jun 05 2013

%F a(n) = - A038608(n). - _Reinhard Zumkeller_, Mar 20 2013

%F a(n+2) = a(n) - 2*(-1)^n. - _G. C. Greubel_, Aug 11 2018

%F a(n) = - A274922(n) if n>0. - _Michael Somos_, Sep 24 2019

%F From _Amiram Eldar_, Oct 24 2023: (Start)

%F Multiplicative with a(2^e) = -2^e, and a(p^e) = p^e for an odd prime p.

%F Dirichlet g.f.: zeta(s-1) * (1-2^(2-s)). (End)

%e G.f. = x - 2*x^2 + 3*x^3 - 4*x^4 + 5*x^5 - 6*x^6 + 7*x^7 - 8*x^8 + 9*x^9 + ...

%p A181983:=n->-(-1)^n * n; seq(A181983(n), n=0..100); # _Wesley Ivan Hurt_, Feb 26 2014

%t a[ n_] := -(-1)^n n;

%t a[ n_] := Sign[n] SeriesCoefficient[ x / (1 + x)^2, {x, 0, Abs @n}];

%t a[ n_] := Sign[n] (Abs @n)! SeriesCoefficient[ x / Exp[ x], {x, 0, Abs @n}];

%t CoefficientList[Series[x/(1+x)^2,{x,0,60}],x] (* or *) LinearRecurrence[{-2,-1},{0,1},60] (* or *) Table[If[OddQ[n],n,-n],{n,0,60}]  (* _Harvey P. Dale_, Apr 22 2022 *)

%o (PARI) {a(n) = -(-1)^n * n};

%o (Haskell)

%o a181983 = negate . a038608

%o a181983_list = [0, 1] ++ map negate

%o    (zipWith (+) a181983_list (map (* 2) $ tail a181983_list))

%o -- _Reinhard Zumkeller_, Mar 20 2013

%o (Magma) [(-1)^(n+1)*n: n in [0..30]]; // _G. C. Greubel_, Aug 11 2018

%Y Cf. A000108, A000169, A001057, A001477, A001787, A002129, A038608, A049347, A154955, A274922.

%K sign,mult,easy

%O 0,3

%A _Michael Somos_, Apr 04 2012