OFFSET

0,1

COMMENTS

From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, nobody gets their own hat.

Also, decimal expansion of cosh(1)-sinh(1). - Mohammad K. Azarian, Aug 15 2006

Also, this is lim_{n->inf} P(n), where P(n) is the probability that a random rooted forest on [n] is a tree. See linked file. - Washington Bomfim, Nov 01 2010

Also, location of the minimum of x^x. - Stanislav Sykora, May 18 2012

Also, -1/e is the global minimum of x*log(x) at x = 1/e and the global minimum of x*e^x at x = -1. - Rick L. Shepherd, Jan 11 2014

Also, the asymptotic probability of success in the secretary problem (also known as the sultan's dowry problem). - Andrey Zabolotskiy, Sep 14 2019

The asymptotic density of numbers with an odd number of trailing zeros in their factorial base representation (A232745). - Amiram Eldar, Feb 26 2021

For large range size s where numbers are chosen randomly r times, the probability when r = s that a number is randomly chosen exactly 1 time. Also the chance that a number was not chosen at all. The general case for the probability of being chosen n times is (r/s)^n / (n! * e^(r/s)). - Mark Andreas, Oct 25 2022

REFERENCES

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.3, p. 14.

Anders Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).

John Harris, Jeffry L. Hirst, and Michael Mossinghoff, Combinatorics and Graph Theory, Springer Science & Business Media, 2009, p. 161.

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (103) on page 20.

Traian Lalescu, Problem 579, Gazeta Matematică, Vol. 6 (1900-1901), p. 148.

John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

Manfred R. Schroeder, Number Theory in Science and Communication, Springer Science & Business Media, 2008, ch. 9.5 Derangements.

Walter D. Wallis and John C. George, Introduction to Combinatorics, CRC Press, 2nd ed. 2016, theorem 5.2 (The Derangement Series).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

Washington Bomfim, Probabilities of connected random forests and derangements, Oct 31 2010.

James Grime and Brady Haran, Derangements, Numberphile video, 2017.

Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619.

Gérard P. Michon, Final Answers: Inclusion-Exclusion.

Michael Penn, A cool, quick limit, YouTube video, 2022.

Eric Weisstein's World of Mathematics, Derangement.

Eric Weisstein's World of Mathematics, Factorial Sums.

Eric Weisstein's World of Mathematics, Spherical Bessel Function of the First Kind.

Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.

Eric Weisstein's World of Mathematics, e.

OEIS Wiki, Number of derangements.

FORMULA

Equals 2*(1/3! + 2/5! + 3/7! + ...). [Jolley]

Equals 1 - Sum_{i >= 1} (-1)^(i - 1)/i!. [Michon]

Equals lim_{x->infinity} (1 - 1/x)^x. - Arkadiusz Wesolowski, Feb 17 2012

Equals j_1(i)/i = cos(i) + i*sin(i), where j_1(z) is the spherical Bessel function of the first kind and i = sqrt(-1). - Stanislav Sykora, Jan 11 2017

Equals Sum_{i>=0} ((-1)^i)/i!. - Maciej Kaniewski, Sep 10 2017

Equals Sum_{i>=0} ((-1)^i)(i^2+1)/i!. - Maciej Kaniewski, Sep 12 2017

From Peter Bala, Oct 23 2019: (Start)

The series representation 1/e = Sum_{k >= 0} (-1)^k/k! is the case n = 0 of the following series acceleration formulas:

1/e = n!*Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*binomial(-x,k) are the row polynomials of A094816. (End)

1/e = 1 - Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n). - Peter Bala, Nov 13 2019

Equals Integral_{x=0..1} x * sinh(x) dx. - Amiram Eldar, Aug 14 2020

Equals lim_{x->oo} (x!)^(1/x)/x. - L. Joris Perrenet, Dec 08 2020

Equals lim_{n->oo} (n+1)!^(1/(n+1)) - n!^(1/n) (Lalescu, 1900-1901). - Amiram Eldar, Mar 29 2022

EXAMPLE

1/e = 0.3678794411714423215955237701614608674458111310317678... = A135005/5.

MATHEMATICA

RealDigits[N[1/E, 6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PROG

CROSSREFS

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Apr 08 2002

EXTENSIONS

More terms from Rick L. Shepherd, Jan 11 2014

STATUS

approved