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

1, 1, 2, 3, 8, 30, 144, 280, 5760, 45360, 44800, 3991680, 43545600, 172972800, 6706022400, 93405312000, 42268262400, 22230464256000, 376610217984000, 250298560512000, 11640679464960000, 196503623737344000, 17841281393295360000

Offset: 0

Author

N. J. A. Sloane, Jan 17 2000

Keywords

• nonn
• frac
• nice
• easy

Comments

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

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

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, Subfactorial

Crossrefs

Cf. A053557 (numerators), A053518-A053520. See also A103816.

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

Programs

Magma

[Denominator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..20]]; // G. C. Greubel, May 16 2019

Mathematica

Table[Denominator[Sum[(-1)^k/k!, {k, 0, n}]], {n, 0, 20}] (* Robert G. Wilson v, Oct 13 2005 *)
Table[ Denominator[1 - Subfactorial[n]/n!], {n, 0, 22}] (* Jean-François Alcover, Feb 11 2014 *)
Denominator[Accumulate[Table[(-1)^k/k!,{k,0,30}]]] (* Harvey P. Dale, Aug 22 2016 *)

PARI

for(n=0,50, print1(denominator(sum(k=0,n,(-1)^k/k!)), ", ")) \\ G. C. Greubel, Nov 05 2017

Python

from math import factorial
from fractions import Fraction
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

Sage

[denominator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..20)] # G. C. Greubel, May 16 2019

Formula

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

Extensions

More terms from Vladeta Jovovic, Mar 31 2000