%I #154 Aug 21 2024 11:21:26

%S 3,6,7,8,7,9,4,4,1,1,7,1,4,4,2,3,2,1,5,9,5,5,2,3,7,7,0,1,6,1,4,6,0,8,

%T 6,7,4,4,5,8,1,1,1,3,1,0,3,1,7,6,7,8,3,4,5,0,7,8,3,6,8,0,1,6,9,7,4,6,

%U 1,4,9,5,7,4,4,8,9,9,8,0,3,3,5,7,1,4,7,2,7,4,3,4,5,9,1,9,6,4,3,7,4,6,6,2,7

%N Decimal expansion of 1/e.

%C 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.

%C Also, decimal expansion of cosh(1)-sinh(1). - _Mohammad K. Azarian_, Aug 15 2006

%C 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

%C Also, location of the minimum of x^x. - _Stanislav Sykora_, May 18 2012

%C 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

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

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

%C 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

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

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

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

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

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

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

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

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

%H G. C. Greubel, <a href="/A068985/b068985.txt">Table of n, a(n) for n = 0..10000</a>

%H Washington Bomfim, <a href="/A068985/a068985.png">Probabilities of connected random forests and derangements</a>, Oct 31 2010.

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=pbXg5EI5t4c">Derangements</a>, Numberphile video, 2017.

%H Peter J. Larcombe, Jack Sutton, and James Stanton, <a href="https://pjm.ppu.edu/sites/default/files/papers/PJM_12%282%29_2023_609_to_619.pdf">A note on the constant 1/e</a>, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619.

%H Gérard P. Michon, <a href="http://www.numericana.com/answer/counting.htm#inex">Final Answers: Inclusion-Exclusion</a>.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=TyOHcs3qD4g">A cool, quick limit</a>, YouTube video, 2022.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Derangement.html">Derangement</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialSums.html">Factorial Sums</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SphericalBesselFunctionoftheFirstKind.html">Spherical Bessel Function of the First Kind</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SultansDowryProblem.html">Sultan's Dowry Problem</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/e.html">e</a>.

%H OEIS Wiki, <a href="/wiki/Number of derangements">Number of derangements</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

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

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

%F Equals lim_{x->infinity} (1 - 1/x)^x. - _Arkadiusz Wesolowski_, Feb 17 2012

%F 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

%F Equals Sum_{i>=0} ((-1)^i)/i!. - _Maciej Kaniewski_, Sep 10 2017

%F Equals Sum_{i>=0} ((-1)^i)(i^2+1)/i!. - _Maciej Kaniewski_, Sep 12 2017

%F From _Peter Bala_, Oct 23 2019: (Start)

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

%F 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)

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

%F Equals Integral_{x=0..1} x * sinh(x) dx. - _Amiram Eldar_, Aug 14 2020

%F Equals lim_{x->oo} (x!)^(1/x)/x. - _L. Joris Perrenet_, Dec 08 2020

%F Equals lim_{n->oo} (n+1)!^(1/(n+1)) - n!^(1/n) (Lalescu, 1900-1901). - _Amiram Eldar_, Mar 29 2022

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

%t RealDigits[N[1/E,6! ]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 18 2009 *)

%o (PARI)

%o default(realprecision, 110);

%o exp(-1) \\ _Rick L. Shepherd_, Jan 11 2014

%Y Cf. A000166, A001113, A068996, A092553, A232745.

%Y Cf. A059193.

%Y Cf. asymptotic probabilities of success for other "nothing but the best" variants of the secretary problem: A325905, A242674, A246665.

%Y Cf. A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

%K nonn,cons,changed

%O 0,1

%A _N. J. A. Sloane_, Apr 08 2002

%E More terms from _Rick L. Shepherd_, Jan 11 2014