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In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt and Thomas Clausen . Specifically, if n is a positive integer and we add 1/ p to the Bernoulli number B 2 n for every prime p such that p − 1 divides 2 n , then we obtain an integer; that is,
The simplest way to compute Gregory coefficients is to use the recurrence formula | | = = | | + + + with G 1 = 1 / 2 . [14] [18] Gregory coefficients may be also computed explicitly via the following differential
The values B n (1) are the Bernoulli numbers B n. Notice that for n ≠ 1 we have = = (), and for n = 1, = = (). The functions P n agree with the Bernoulli polynomials on the interval [0, 1] and are periodic with period 1.
The Bernoulli numbers have various definitions (see Bernoulli number#Definitions), such as that they are the coefficients of the exponential generating function = ( +) = =!. Then Faulhaber's formula is that ∑ k = 1 n k p = 1 p + 1 ∑ k = 0 p ( p + 1 k ) B k n p − k + 1 . {\displaystyle \sum _{k=1}^{n}k^{p}={\frac {1}{p+1}}\sum _{k=0}^{p ...
The categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values. The Beta distribution is the conjugate prior of the Bernoulli distribution. [5] The geometric distribution models the number of independent and identical Bernoulli trials needed to get one success.
On a note more distantly related to combinatorics, the second section also discusses the general formula for sums of integer powers; the free coefficients of this formula are therefore called the Bernoulli numbers, which influenced Abraham de Moivre's work later, [16] and which have proven to have numerous applications in number theory. [22]
The prime number 2 is often considered regular as well. The class number of the cyclotomic field is the number of ideals of the ring of integers Z(ζ p) up to equivalence. Two ideals I, J are considered equivalent if there is a nonzero u in Q(ζ p) so that I = uJ. The first few of these class numbers are listed in OEIS: A000927.