%I A073003
%S A073003 5,9,6,3,4,7,3,6,2,3,2,3,1,9,4,0,7,4,3,4,1,0,7,8,4,9,9,3,6,9,2,7,9,3,7,
%T A073003 6,0,7,4,1,7,7,8,6,0,1,5,2,5,4,8,7,8,1,5,7,3,4,8,4,9,1,0,4,8,2,3,2,7,2,
%U A073003 1,9,1,1,4,8,7,4,4,1,7,4,7,0,4,3,0,4,9,7,0,9,3,6,1,2,7,6,0,3,4,4,2,3,7
%N A073003 Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant.
%C A073003 0! - 1! + 2! - 3! + 4! - 5! + ... = (Borel) sum_{n=0}^infinity (-y)^n 
               n! = KummerU(1,1,1/y)/y
%C A073003 Decimal expansion of phi(1) where phi(x)=integral(t=0,infinity, e^-t/
               (x+t) dt ). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 11 2003
%C A073003 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%C A073003 The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... 
               , m=>-1, is intimately related to Gompertz's constant. We discovered 
               that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with 
               A000110 the Bell numbers and A040027 a sequence that was published 
               by Gould, see for more information A163940.
%C A073003 (End)
%D A073003 Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 171
%D A073003 Bruce C. Berndt, Ramanujan's notebooks Part I, Springer, p. 144-145.
%D A073003 Walls, H. S., Analytic Theory of Continued Fractions, Van Nostrand, New 
               York, 1948, p. 356
%H A073003 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/
               sci/math/MiscellaneousMathematicalConstants/chap20.html">-exp(1)*Ei(-1)</
               a>
%H A073003 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GompertzConstant.html">Link to a section of The World of Mathematics</
               a>
%H A073003 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ExponentialIntegral.html">Exponential Integral</a>
%H A073003 A. I. Aptekarev, <a href="http://arxiv.org/abs/0902.1768">On linear forms 
               containing the Euler constant</a>, arXiv:0902.1768 [math.NT]. [From 
               R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 14 2009]
%H A073003 G.H. Hardy, <a href="http://www.archive.org/texts/flipbook/flippy.php?id=divergentseries033523mbp">
               Divergent Series </a>, Oxford University Press, 1949. p. 29. [From 
               Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009]
%H A073003 Ed Sandifer, <a href="http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2032%20divergent%20series.pdf\
               ">Divergent Series</a>, How Euler Did It, MAA Online, June 2006. 
               [From Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009]
%F A073003 phi(1)=e*(sum(k>=1, (-1)^(k-1)/k/k!) - Gamma)=0.596347362323194... where 
               Gamma is the Euler constant.
%F A073003 G = 0.596347... = 1/(1+1/(1+1/(1+2/(1+2/(1+3/(1+3/(1+4/(1+4/(1+5/(1+5/
               (1+6/(...- Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 14 2005
%e A073003 0.59634736232319407434...
%t A073003 RealDigits[ N[ -Exp[1]*ExpIntegralEi[ -1], 110]] [[1]]
%Y A073003 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%Y A073003 Equals A001113*A099285.
%Y A073003 (End)
%Y A073003 Sequence in context: A134879 A051158 A117605 this_sequence A087498 A121060 
               A021630
%Y A073003 Adjacent sequences: A073000 A073001 A073002 this_sequence A073004 A073005 
               A073006
%K A073003 cons,nonn
%O A073003 0,1
%A A073003 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 03 2002
%E A073003 Additional references from Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Oct 10 2005
%E A073003 Link corrected by Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 01 
               2009