OFFSET
0,3
COMMENTS
From Harry Richman, Aug 13 2024: (Start)
Euler argued this sequence converges to 0.596347... (A073003 = Gompertz's constant); see Lagarias Section 2.5.
This sequence converges in the p-adic topology, for every prime number p. (End)
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., Vol. 50, No. 4 (2013), pp. 527-628; preprint, arXiv:1303.1856 [math.NT], 2013.
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
KEYWORD
easy,sign
AUTHOR
Henry Bottomley, Nov 13 2000
EXTENSIONS
Corrections and more information from Michael Somos, Feb 19 2003
STATUS
approved