A058006 Alternating factorials: 0! - 1! + 2! - ... + (-1)^n n!

1, 0, 2, -4, 20, -100, 620, -4420, 35900, -326980, 3301820, -36614980, 442386620, -5784634180, 81393657020, -1226280710980, 19696509177020, -335990918918980, 6066382786809020, -115578717622022980, 2317323290554617020, -48773618881154822980

Offset

0,3

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..400

Eric Weisstein's MathWorld, Incomplete Gamma Function.

Formula

a(n) = (-1)^n n! + a(n-1) = A005165(n)(-1)^n + 1.

a(n) = -(n-1)*a(n-1) + n*a(n-2), n>0.

E.g.f.: d/dx ((GAMMA(0,1)-GAMMA(0,1+x))*exp(1+x)). - Max Alekseyev, Jul 05 2010

G.f.: G(0)/(1-x), where G(k)= 1 - (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

0 = a(n)*(-a(n+1) + a(n+3)) + a(n+1)*(2*a(n+1) - 2*a(n+2) -a(n+3)) + a(n+2)*(a(n+2)) if n>=-1. - Michael Somos, Jan 28 2014

a(n) = exp(1)*Gamma(0,1) + (-1)^n*exp(1)*(n+1)!*Gamma(-n-1,1), where Gamma(a,x) is the upper incomplete Gamma function. - Vladimir Reshetnikov, Oct 29 2015

Example

a(5) = 0!-1!+2!-3!+4!-5! = 1-1+2-6+24-120 = -100.
G.f. = 1 + 2*x^2 - 4*x^3 + 20*x^4 - 100*x^5 + 620*x^6 - 4420*x^7 + 35900*x^8 + ...

Mathematica

a[ n_] := Sum[ (-1)^k k!, {k, 0, n}]; (* Michael Somos, Jan 28 2014 *)

Prog

(PARI) {a(n) = sum(k=0, n, (-1)^k * k!)}; /* Michael Somos, Jan 28 2014 */
(Haskell)
a058006 n = a058006_list !! n
a058006_list = scanl1 (+) a133942_list
-- Reinhard Zumkeller, Mar 02 2014

Crossrefs

Cf. A000142, A003422, A005165, A153229 (absolute values), A136580.

Partial sums of A133942.

Sequence in context: A108879 A341855 A337038 * A153229 A325617 A013329

Adjacent sequences: A058003 A058004 A058005 * A058007 A058008 A058009

Keyword

easy,sign

Author

Henry Bottomley, Nov 13 2000

Extensions

Corrections and more information from Michael Somos, Feb 19 2003

Status

approved