A024000 a(n) = 1 - n.

1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30, -31, -32, -33, -34, -35, -36, -37, -38, -39, -40, -41, -42, -43, -44, -45, -46, -47, -48, -49, -50, -51, -52, -53, -54, -55, -56

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

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]

Examples

			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
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A000007, A000166, A087981, A137775. - Michael Somos, Jan 19 2011

Cf. A001489, A094816.

A022958 shifted left.

Programs

Magma

[1-n: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Maple

A024000:=n->1-n: seq(A024000(n), n=0..100); # Wesley Ivan Hurt, Mar 02 2016

Mathematica

CoefficientList[Series[(1 - 2 x)/(1 - x)^2, {x, 0, 60}], x] Range[0, 60]!
CoefficientList[Series[Exp[x] (1 - x), {x, 0, 60}], x]
1-Range[0,60] (* Harvey P. Dale, Sep 18 2013 *)
Flatten[NestList[(#/.x_/;x>1->Sequence[x,2x])-1&,{1},60]]
(* Robert G. Wilson v, Mar 02 2016 *)

PARI

{a(n) = 1 - n} /* Michael Somos, Jan 19 2011 */

Formula

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

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

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

a(n) = 2*a(n-1)-a(n-2) for n>1. - Wesley Ivan Hurt, Mar 02 2016