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A053556 Denominator of Sum_{k=0..n} (-1)^k/k!.

%I A053556 #52 Feb 16 2025 08:32:42
%S A053556 1,1,2,3,8,30,144,280,5760,45360,44800,3991680,43545600,172972800,
%T A053556 6706022400,93405312000,42268262400,22230464256000,376610217984000,
%U A053556 250298560512000,11640679464960000,196503623737344000,17841281393295360000
%N A053556 Denominator of Sum_{k=0..n} (-1)^k/k!.
%C A053556 Denominator of probability of a derangement of n things (A000166(n)/n!).
%C A053556 Also numerators of successive convergents to e using continued fraction 2 +1/(1 +1/(2 +2/(3 +3/(4 +4/(5 +5/(6 +6/(7 +7/8 +...))))))).
%D A053556 L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
%D A053556 E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.
%H A053556 G. C. Greubel, <a href="/A053556/b053556.txt">Table of n, a(n) for n = 0..450</a> (terms 0..100 from T. D. Noe)
%H A053556 Leonhardo Eulero, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k69587">Introductio in analysin infinitorum. Tomus primus</a>, Lausanne, 1748.
%H A053556 L. Euler, Introduction à l'analyse infinitésimale, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k3884z">Tome premier</a>, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k38858">Tome second</a>, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
%H A053556 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Subfactorial.html">Subfactorial</a>
%F A053556 Let exp(-x)/(1-x) = Sum_{n>=0} (a_n/b_n) * x^n. Then sequence b_n is A053556. - _Aleksandar Petojevic_, Apr 14 2004
%e A053556 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...
%t A053556 Table[Denominator[Sum[(-1)^k/k!, {k, 0, n}]], {n, 0, 20}] (* _Robert G. Wilson v_, Oct 13 2005 *)
%t A053556 Table[ Denominator[1 - Subfactorial[n]/n!], {n, 0, 22}] (* _Jean-François Alcover_, Feb 11 2014 *)
%t A053556 Denominator[Accumulate[Table[(-1)^k/k!,{k,0,30}]]] (* _Harvey P. Dale_, Aug 22 2016 *)
%o A053556 (PARI) for(n=0,50, print1(denominator(sum(k=0,n,(-1)^k/k!)), ", ")) \\ _G. C. Greubel_, Nov 05 2017
%o A053556 (Magma) [Denominator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..20]]; // _G. C. Greubel_, May 16 2019
%o A053556 (Sage) [denominator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..20)] # _G. C. Greubel_, May 16 2019
%o A053556 (Python)
%o A053556 from math import factorial
%o A053556 from fractions import Fraction
%o A053556 def A053556(n): return sum(Fraction(-1 if k&1 else 1,factorial(k)) for k in range(n+1)).denominator # _Chai Wah Wu_, Jul 31 2023
%Y A053556 Cf. A053557 (numerators), A053518-A053520. See also A103816.
%Y A053556 a(n) = (D(n, n) of A103360), A053557/A053556 = A000166/n! = (N(n, n) of A103361)/(D(n, n) of A103360).
%K A053556 nonn,frac,nice,easy
%O A053556 0,3
%A A053556 _N. J. A. Sloane_, Jan 17 2000
%E A053556 More terms from _Vladeta Jovovic_, Mar 31 2000