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%I #21 Oct 29 2015 22:38:18
%S 1,0,2,-4,20,-100,620,-4420,35900,-326980,3301820,-36614980,442386620,
%T -5784634180,81393657020,-1226280710980,19696509177020,
%U -335990918918980,6066382786809020,-115578717622022980,2317323290554617020,-48773618881154822980
%N Alternating factorials: 0! - 1! + 2! - ... + (-1)^n n!
%H Reinhard Zumkeller, <a href="/A058006/b058006.txt">Table of n, a(n) for n = 0..400</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.
%F a(n) = (-1)^n n! + a(n-1) = A005165(n)(-1)^n + 1.
%F a(n) = -(n-1)*a(n-1) + n*a(n-2), n>0.
%F E.g.f.: d/dx ((GAMMA(0,1)-GAMMA(0,1+x))*exp(1+x)). - _Max Alekseyev_, Jul 05 2010
%F 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
%F 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
%F 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
%e a(5) = 0!-1!+2!-3!+4!-5! = 1-1+2-6+24-120 = -100.
%e G.f. = 1 + 2*x^2 - 4*x^3 + 20*x^4 - 100*x^5 + 620*x^6 - 4420*x^7 + 35900*x^8 + ...
%t a[ n_] := Sum[ (-1)^k k!, {k, 0, n}]; (* _Michael Somos_, Jan 28 2014 *)
%o (PARI) {a(n) = sum(k=0, n, (-1)^k * k!)}; /* _Michael Somos_, Jan 28 2014 */
%o (Haskell)
%o a058006 n = a058006_list !! n
%o a058006_list = scanl1 (+) a133942_list
%o -- _Reinhard Zumkeller_, Mar 02 2014
%Y Cf. A000142, A003422, A005165, A153229 (absolute values), A136580.
%Y Partial sums of A133942.
%K easy,sign
%O 0,3
%A _Henry Bottomley_, Nov 13 2000
%E Corrections and more information from _Michael Somos_, Feb 19 2003
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