%I A133942
%S A133942 1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,
%T A133942 6227020800,87178291200,1307674368000,20922789888000,355687428096000,
%U A133942 6402373705728000,121645100408832000,2432902008176640000
%V A133942 1,-1,2,-6,24,-120,720,-5040,40320,-362880,3628800,-39916800,479001600,
               -6227020800,
%W A133942 87178291200,-1307674368000,20922789888000,-355687428096000,6402373705728000,
%X A133942 -121645100408832000,2432902008176640000
%N A133942 (-1)^n * n!.
%C A133942 A variant of A000142, the factorial numbers. - N. J. A. Sloane (njas(AT)research.att.com), 
               Oct 03 2007
%C A133942 The terms of this sequences form the factorial series which Euler called 
               the divergent series par excellence.
%C A133942 Euler summed this series to 0.596347... (A073003 = Gompertz's constant).
%C A133942 Sum_{n>=0} 1/a(n) = 1/e [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Mar 03 2009]
%D A133942 V. S. Varadarajan, Euler and his Work on Infinite Series, Bull. Amer. 
               Math. Soc., 44 (No. 4, 2007), 515-539. (See p. 527 and 530.)
%F A133942 Sum((-1)^i * i^n * binomial(n, i), i=0..n) = (-1)^n * n! - Yong Kong 
               (ykong(AT)curagen.com), Dec 26 2000
%F A133942 Stirling transform of a(n)=[1,-1,2,-6,24,...] is A000007(n)=[1,0,0,0,
               0,...].
%F A133942 a(n) = -n * a(n-1) unless n=0.
%F A133942 E.g.f.: 1/(1 + x).
%F A133942 G.f.: integral(t=1/x,infinity, (e^-t)/t) e^(1/x)/x = 1/(1 + x/(1 + x/
               (1 + 2x/(1 + 2x/(1 + 3x/(1 + 3x/(1 + ...))))))).
%o A133942 (PARI) {a(n) = if( n<0, 0, (-1)^n * n! )}
%Y A133942 Cf. (-1)^n * A000142(n) = a(n).
%Y A133942 Sequence in context: A000142 A104150 A124355 this_sequence A159333 A165233 
               A074166
%Y A133942 Adjacent sequences: A133939 A133940 A133941 this_sequence A133943 A133944 
               A133945
%K A133942 sign
%O A133942 0,3
%A A133942 Michael Somos, Sep 30 2007