%I #33 Jul 31 2023 14:17:35
%S 0,1,1,2,5,19,91,177,3641,28673,28319,2523223,27526069,109339663,
%T 4239014627,59043418019,26718637649,14052333488521,238063061452591,
%U 158218865944829,7358312808534631,124213980448686521,11277840764547411113,67527236643922308689
%N Numerator of Sum_{k=1..n} (-1)^(k+1)/k!.
%C Numerator of (n! - A000166(n))/n!.
%C Numerator of 1 - A053557/A053556.
%H Alois P. Heinz, <a href="/A103816/b103816.txt">Table of n, a(n) for n = 0..250</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ContinuedFractionConstants.html">Continued Fraction Constants</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedContinuedFraction.html">Generalized Continued Fraction</a>
%F The Aitken delta-squared process leaves the sequence S(n) = Sum_{k=1..n} (-1)^(k+1)/k! essentially unchanged: S(n+3) = (S(n)*S(n+2) - (S(n+1))^2)/(S(n) + S(n+2) - 2*S(n+1)).
%F Numerators of coefficients in expansion of (1 - exp(-x)) / (1 - x). - _Ilya Gutkovskiy_, May 24 2022
%e 0, 1, 1/2, 2/3, 5/8, 19/30, 91/144, 177/280, 3641/5760, 28673/45360, 28319/44800, ...
%p b:= proc(n) b(n):=`if`(n<2, 1-n, (n-1)*(b(n-1)+b(n-2))) end:
%p a:= n-> numer((n!-b(n))/n!):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 15 2013
%t Table[Numerator[Sum[ -(-1)^k/k!, {k, n}]], {n, 0, 22}] (* _Robert G. Wilson v_ *)
%t Table[Numerator[1 - Subfactorial[n]/n!], {n, 0, 23}] (* _Jean-François Alcover_, Feb 11 2014 *)
%t Join[{0},Accumulate[Times@@@Partition[Riffle[1/Range[30]!,{1,-1},{2,-1,2}],2]]//Numerator] (* _Harvey P. Dale_, Apr 18 2023 *)
%o (Python)
%o from math import factorial
%o from fractions import Fraction
%o def A103816(n): return sum(Fraction(1 if k&1 else -1,factorial(k)) for k in range(1,n+1)).numerator # _Chai Wah Wu_, Jul 31 2023
%Y Cf. A053556 (denominators).
%K nonn,frac,nice,easy
%O 0,4
%A _N. J. A. Sloane_, Apr 02 2005
%E More terms from _Robert G. Wilson v_, Oct 13 2005