|
%I #57 Nov 10 2023 12:27:47
%S 1,0,1,1,3,11,53,103,2119,16687,16481,1468457,16019531,63633137,
%T 2467007773,34361893981,15549624751,8178130767479,138547156531409,
%U 92079694567171,4282366656425369,72289643288657479,6563440628747948887,39299278806015611311
%N Numerator of Sum_{k=0..n} (-1)^k/k!.
%C Numerator of probability of a derangement of n things (A000166(n)/n! or !n/n!).
%C Also denominators 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 L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
%D E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.
%H G. C. Greubel, <a href="/A053557/b053557.txt">Table of n, a(n) for n = 0..450</a> (terms 0..100 from T. D. Noe)
%H Leonhardo Eulero, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k69587">Introductio in analysin infinitorum. Tomus primus</a>, Lausanne, 1748.
%H 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 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>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Subfactorial.html">Subfactorial</a>
%F Let exp(-x)/(1-x) = Sum_{n >= 0} (a_n/b_n)*x^n. Then sequence a_n is A053557. - _Aleksandar Petojevic_, Apr 14 2004
%e 1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...
%t Numerator[CoefficientList[Series[Exp[-x]/(1-x), {x, 0, 30}], x]] (* _Jean-François Alcover_, Nov 18 2011 *)
%t Table[Numerator[Sum[(-1)^k/k!,{k,0,n}]],{n,0,30}] (* _Harvey P. Dale_, Dec 02 2011 *)
%t Join[{1, 0}, Numerator[RecurrenceTable[{a[n]==a[n-1]+a[n-2]/(n-2), a[1] ==0, a[2]==1}, a, {n,2,30}]]] (* _Terry D. Grant_, May 07 2017; corrected by _G. C. Greubel_, May 16 2019 *)
%o (PARI) for(n=0, 30, print1(numerator(sum(k=0,n, (-1)^k/k!)), ", ")) \\ _G. C. Greubel_, Nov 05 2017
%o (Magma) [Numerator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..30]]; // _G. C. Greubel_, May 16 2019
%o (Sage) [numerator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..30)] # _G. C. Greubel_, May 16 2019
%o (Python)
%o from fractions import Fraction
%o from math import factorial
%o def A053557(n): return sum(Fraction(-1 if k&1 else 1,factorial(k)) for k in range(n+1)).numerator # _Chai Wah Wu_, Jul 31 2023
%Y Cf. A000166/A000142, A053556 (denominators), A053518-A053520. See also A103816.
%Y a(n) = (N(n, n) of A103361), A053557/A053556 = A000166/n! = (N(n, n) of A103361)/(D(n, n) of A103360), Cf. A053518-A053520.
%K nonn,frac,nice,easy
%O 0,5
%A _N. J. A. Sloane_, Jan 17 2000
%E More terms from _Vladeta Jovovic_, Mar 31 2000
|
