A325905 -id:A325905 - cp's OEIS frontend

A068985 Decimal expansion of 1/e.

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, 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, 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

Offset: 0

N. J. A. Sloane, Apr 08 2002

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

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

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Sections 1.3 and 5,23,3, pp. 14, 409.
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.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 26, page 233.
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.

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.
Index entries for transcendental numbers.

Cf. A000166, A001113, A068996, A092553, A232745.
Cf. A059193.
Cf. asymptotic probabilities of success for other "nothing but the best" variants of the secretary problem: A325905, A242674, A246665.
Cf. A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

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

default(realprecision, 110);
exp(-1) \\ Rick L. Shepherd, Jan 11 2014

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

More terms from Rick L. Shepherd, Jan 11 2014

A242674 Decimal expansion of the asymptotic probability of success in one of the Secretary problems.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 20 2014

This is the asymptotic probability of success for the full-information problem with uniform distribution: we can not only determine which of any two applicants is better than the other, but also determine his/her absolute value, and that value is known to be uniformly distributed on a known interval (say, [0, 1]), independently for each applicant; so we have more information than in the basic version of the problem (for which the chance of success is given by A068985), so the chance of success is greater. Here the number of applicants is known in advance (although we consider the limiting case when it is sent to infinity); for the variant where it is itself a random variable, see A325905. - Andrey Zabolotskiy, Sep 14 2019

			0.580164223920855346426008323572997276633...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.

Eric Weisstein's MathWorld, Sultan's Dowry Problem
Wikipedia, Secretary problem

Cf. A054404, A242672, A242673, A068985, A325905.

a = x /. FindRoot[ExpIntegralEi[x] - EulerGamma - Log[x] == 1, {x, 1}, WorkingPrecision -> 105]; Exp[-a] - (Exp[a]-a-1)*ExpIntegralEi[-a] // RealDigits[#, 10, 100]& // First

exp(-a) - (exp(a)-a-1)*Ei(-a), where a is the unique real solution of the equation Ei(a)-gamma-log(a) = 1, Ei being the exponential integral function, and gamma the Euler-Mascheroni constant (0.5772156649...).

A246665 Decimal expansion of the asymptotic probability of success in the full-information secretary problem with uniform distribution when the number of applicants is also uniformly distributed.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 01 2014

In this variant of the secretary problem, the applicants' values are independently distributed on a known interval, like in A242674; and the number of applicants is itself a random variable with uniform distribution on 1..n (and then the limit n -> infinity is taken), like in A325905. So we have more information than in the variant considered in A325905 but less information than in the variant considered in A242674. Hence A325905 < this constant < A242674. - Andrey Zabolotskiy, Sep 14 2019

			0.43517080558012765805918991284785841042796259475347024702979123...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15 Optimal stopping constants, p. 361.

Steven R. Finch, Errata and Addenda to Mathematical Constants, Sec. 5.15 Optimal Stopping Constants, pp. 53-55, arXiv:2001.00578 [math.HO], 2020.
Zdzisław Porosiński, On best choice problems having similar solutions, Statistics & Probability Letters, 56 (2002), 321-327.
Eric Weisstein's MathWorld, Sultan's Dowry Problem
Wikipedia, Secretary problem

Cf. A246664, A068985, A325905, A242674.

a = x /. FindRoot[E^x*(1 - EulerGamma - Log[x] + ExpIntegralEi[-x]) - (EulerGamma + Log[x] - ExpIntegralEi[x]) == 1, {x, 2}, WorkingPrecision -> 102]; (1 - E^a)*ExpIntegralEi[-a] - (E^-a + a*ExpIntegralEi[-a])*(EulerGamma + Log[a] - ExpIntegralEi[a]) // RealDigits // First

(1 - e^a)*Ei(-a) - (e^(-a) + a*Ei(-a))*(gamma + log(a) - Ei(a)), where a is A246664, gamma is Euler's constant and Ei is the exponential integral function.

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A068985 Decimal expansion of 1/e.

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A242674 Decimal expansion of the asymptotic probability of success in one of the Secretary problems.

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A246665 Decimal expansion of the asymptotic probability of success in the full-information secretary problem with uniform distribution when the number of applicants is also uniformly distributed.

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