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a(n) = 1 - n.
15

%I #33 Sep 08 2022 08:44:48

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

%T -19,-20,-21,-22,-23,-24,-25,-26,-27,-28,-29,-30,-31,-32,-33,-34,-35,

%U -36,-37,-38,-39,-40,-41,-42,-43,-44,-45,-46,-47,-48,-49,-50,-51,-52,-53,-54,-55,-56

%N a(n) = 1 - n.

%C a(n) is the weighted sum over all derangements (permutations with no fixed points) of n elements where each permutation with an odd number of cycles has weight +1 and each with an even number of cycles has weight -1. [_Michael Somos_, Jan 19 2011]

%H Vincenzo Librandi, <a href="/A024000/b024000.txt">Table of n, a(n) for n = 0..1000</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

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

%F E.g.f.: (1-x)*exp(x).

%F a(n) = Sum_{k=0..n} A094816(n,k)*(-1)^k (alternating row sums of Poisson-Charlier coefficient matrix).

%F O.g.f.: (1-2*x)/(1-x)^2. a(n+1) = A001489(n). - _R. J. Mathar_, May 28 2008

%F a(n) = 2*a(n-1)-a(n-2) for n>1. - _Wesley Ivan Hurt_, Mar 02 2016

%e a(4) = -3 because there are 6 derangements with one 4-cycle with weight -1 and 3 derangements with two 2-cycles with weight +1. - _Michael Somos_, Jan 19 2011

%p A024000:=n->1-n: seq(A024000(n), n=0..100); # _Wesley Ivan Hurt_, Mar 02 2016

%t CoefficientList[Series[(1 - 2 x)/(1 - x)^2, {x, 0, 60}], x] Range[0, 60]!

%t CoefficientList[Series[Exp[x] (1 - x), {x, 0, 60}], x]

%t 1-Range[0,60] (* _Harvey P. Dale_, Sep 18 2013 *)

%t Flatten[NestList[(#/.x_/;x>1->Sequence[x,2x])-1&,{1},60]]

%t (* _Robert G. Wilson v_, Mar 02 2016 *)

%o (PARI) {a(n) = 1 - n} /* _Michael Somos_, Jan 19 2011 */

%o (Magma) [1-n: n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011

%Y Cf. A000007, A000166, A087981, A137775. - _Michael Somos_, Jan 19 2011

%Y Cf. A001489, A094816.

%K sign,easy

%O 0,4

%A _N. J. A. Sloane_