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Bernoulli number

The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the exponential generating function(1)These numbers arise in the series expansions of trigonometric functions, and areextremely important in number theory and analysis.There are actually two definitions for the Bernoulli numbers. To distinguish them, the Bernoulli numbers as defined in modern usage (National Institute of Standards and Technology convention) are written , while the Bernoulli numbers encountered in older literature are written (Gradshteyn and Ryzhik 2000). In each case, the Bernoulli numbers are a special case of the Bernoulli polynomials or with and .The Bernoulli number and polynomial should not be confused with the Bell numbers and Bell polynomial, which are also commonly denoted and , respectively.Bernoulli numbers defined by the modern definition are denoted and sometimes called "even-index" Bernoulli numbers...

Khinchin's constant

Let(1)be the simple continued fraction of a "generic" real number , where the numbers are the partial quotients. Khinchin (1934) considered the limit of the geometric mean(2)as . Amazingly, except for a set of measure 0, this limit is a constant independent of given by(3)(OEIS A002210), as proved in Kac (1959).The constant is known as Khinchin's constant, and is commonly also spelled "Khintchine'sconstant" (Shanks and Wrench 1959, Bailey et al. 1997).It is implemented as Khinchin, where its value is cached to 1100-digit precision. However, the numerical value of is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower.It is not known if is irrational, let alone transcendental.While it is known that almost all numbers have limits that approach , this fact has not been proven for any explicit real number , e.g., a real number cast in terms of fundamental constants..

Subfactorial

The th subfactorial (also called the derangement number; Goulden and Jackson 1983, p. 48; Graham et al. 2003, p. 1050) is the number of permutations of objects in which no object appears in its natural place (i.e., "derangements").The term "subfactorial "was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1809) calculated the first ten terms.The first few values of for , 2, ... are 0, 1, 2, 9, 44, 265, 1854, 14833, ... (OEIS A000166). For example, the only derangements of are and , so . Similarly, the derangements of are , , , , , , , , and , so .Sums and formulas for include(1)(2)(3)(4)where is a factorial, is a binomial coefficient, and is the incomplete gamma function.Subfactorials are implemented in the WolframLanguage as Subfactorial[n].A plot the real and imaginary parts of the subfactorial generalized to any real argument is illustrated above, with the usual integer-valued subfactorial..

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