A053557 Numerator of Sum_{k=0..n} (-1)^k/k!.

1, 0, 1, 1, 3, 11, 53, 103, 2119, 16687, 16481, 1468457, 16019531, 63633137, 2467007773, 34361893981, 15549624751, 8178130767479, 138547156531409, 92079694567171, 4282366656425369, 72289643288657479, 6563440628747948887, 39299278806015611311

Offset: 0

Author

N. J. A. Sloane, Jan 17 2000

Keywords

• nonn
• frac
• nice
• easy

Comments

Numerator of probability of a derangement of n things (A000166(n)/n! or !n/n!).

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 + ...)))))))).

Examples

			1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...
		

References

• L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
• E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..450 (terms 0..100 from T. D. Noe)
• Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
• L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
• Eric Weisstein's World of Mathematics, Continued Fraction Constants
• Eric Weisstein's World of Mathematics, Generalized Continued Fraction
• Eric Weisstein's World of Mathematics, Subfactorial

Crossrefs

Cf. A000166/A000142, A053556 (denominators), A053518-A053520. See also A103816.

a(n) = (N(n, n) of A103361), A053557/A053556 = A000166/n! = (N(n, n) of A103361)/(D(n, n) of A103360), Cf. A053518-A053520.

Programs

Magma

[Numerator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, May 16 2019

Mathematica

Numerator[CoefficientList[Series[Exp[-x]/(1-x), {x, 0, 30}], x]] (* Jean-François Alcover, Nov 18 2011 *)
Table[Numerator[Sum[(-1)^k/k!,{k,0,n}]],{n,0,30}] (* Harvey P. Dale, Dec 02 2011 *)
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 *)

PARI

for(n=0, 30, print1(numerator(sum(k=0,n, (-1)^k/k!)), ", ")) \\ G. C. Greubel, Nov 05 2017

Python

from fractions import Fraction
from math import factorial
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

Sage

[numerator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..30)] # G. C. Greubel, May 16 2019

Formula

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

Extensions

More terms from Vladeta Jovovic, Mar 31 2000