cp's OEIS Frontend

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

Also, decimal expansion of the real part of e^i. - Bruno Berselli, Feb 08 2013

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

Previous name was: Decimal expansion of the constant 1 + 1/3! + 1/6! + 1/9! + ... = 1.16805 83133 75918 ... .

Define a sequence R(n) of real numbers by R(n) := Sum_{k>=0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(0); the decimal expansions of R(2) - R(1) = 1/1! + 1/4! + 1/7! and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143820 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i=0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.

Bowman and Mc Laughlin (Corollary 10 with m = -1) give a continued fraction expansion for this constant and deduce the constant is irrational. - Peter Bala, Apr 17 2017

Define a sequence R(n) of real numbers by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(2) - R(1); the decimal expansions of R(0) = 1 + 1/3! + 1/6! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143819 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1).

R(n) as a linear combination of R(0), R(1) and R(2) - R(1).

========================================

| linear combination of

R(n) | R(0) R(1) R(2) - R(1)

========================================

R(3) | 1 1 3

R(4) | 6 2 7

R(5) | 25 11 16

R(6) | 91 66 46

R(7) | 322 352 203

R(8) | 1232 1730 1178

R(9) | 5672 8233 7242

R(10) | 32202 39987 43786

...

The column entries are from A143815, A143816 and A143817.

The Abraham Ungar 1982 article defines H_{n,r}(z) = Sum_{k>=0} z^(nk+r)/(nk+r)! as equation (1). The constant is H_{3,1}(1). In equation (13) H_{3,1}(x) = (exp(x) + 2 * exp(-x/2) * cos(sqrt(3)/2*x - 2*Pi/3))/3. In equation (12) the expression H_{3,1}(x) = (e^x + q_2 e^{q_1 x} + q_1 e^{q_2 x})/3 where q_1 = (-1 + I sqrt(3))/2 and q_2 = (-1 - I sqrt(3))/2 is given for H_{3,2}(x) instead. - Michael Somos, Nov 01 2024

For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity.