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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Euler numbers are a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
''En'' of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s defined by the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion :\frac = \frac = \sum_^\infty \frac \cdot t^n, where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely: :E_n=2^nE_n(\tfrac 12). The Euler numbers appear in the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansions of the secant and
hyperbolic secant In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
functions. The latter is the function in the definition. They also occur in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, specifically when counting the number of
alternating permutation In combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set is a permutation (arrangement) of those numbers so that each entry is alternately greater or less than the preceding entry. For example, the five alte ...
s of a set with an even number of elements.


Examples

The odd-indexed Euler numbers are all
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
. The even-indexed ones have alternating signs. Some values are: : Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive . This article adheres to the convention adopted above.


Explicit formulas


In terms of Stirling numbers of the second kind

Following two formulas express the Euler numbers in terms of
Stirling numbers of the second kind In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
: E_=2^\sum_^\frac\left(3\left(\frac\right)^-\left(\frac\right)^\right), : E_=-4^\sum_^(-1)^\cdot \frac\cdot \left(\frac\right)^, where S(n,\ell) denotes the
Stirling numbers of the second kind In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
, and x^=(x)(x+1)\cdots (x+\ell-1) denotes the
rising factorial In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...
.


As a double sum

Following two formulas express the Euler numbers as double sums :E_=(2 n+1)\sum_^ (-1)^\frac\binom\sum _^\binom(2q-\ell)^, :E_=\sum_^(-1)^ \frac\sum_^(-1)^ \binom(k-\ell)^.


As an iterated sum

An explicit formula for Euler numbers is: :E_=i\sum _^ \sum _^k \binom\frac, where denotes the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
with .


As a sum over partitions

The Euler number can be expressed as a sum over the even
partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
of , : E_ = (2n)! \sum_ \binom K \delta_ \left( -\frac \right)^ \left( -\frac \right)^ \cdots \left( -\frac \right)^ , as well as a sum over the odd partitions of , : E_ = (-1)^ (2n-1)! \sum_ \binom K \delta_ \left( -\frac \right)^ \left( \frac \right)^ \cdots \left( \frac \right)^ , where in both cases and : \binom K \equiv \frac is a
multinomial coefficient In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer an ...
. The
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
s in the above formulas restrict the sums over the s to and to , respectively. As an example, : \begin E_ & = 10! \left( - \frac + \frac + \frac - \frac- \frac +\frac - \frac\right) \\ pt& = 9! \left( - \frac + \frac + \frac +\frac- \frac -\frac + \frac - \frac\right) \\ pt& = -50\,521. \end


As a determinant

is given by the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
: \begin E_ &=(-1)^n (2n)!~ \begin \frac& 1 &~& ~&~\\ \frac& \frac & 1 &~&~\\ \vdots & ~ & \ddots~~ &\ddots~~ & ~\\ \frac& \frac& ~&\frac & 1\\ \frac&\frac& \cdots & \frac & \frac\end. \end


As an integral

is also given by the following integrals: : \begin (-1)^n E_ & = \int_0^\infty \frac\; dt =\left(\frac2\pi\right)^ \int_0^\infty \frac\; dx\\ pt&=\left(\frac2\pi\right)^ \int_0^1\log^\left(\tan \frac \right)\,dt =\left(\frac2\pi\right)^\int_0^ \log^\left(\tan \frac \right)\,dx\\ pt&= \frac \int_0^ x \log^ (\tan x)\,dx = \left(\frac2\pi\right)^ \int_0^\pi \frac \log^ \left(\tan \frac \right)\,dx.\end


Congruences

W. Zhang obtained the following combinational identities concerning the Euler numbers, for any prime p , we have : (-1)^ E_ \equiv \textstyle\begin 0 \mod p &\textp\equiv 1\bmod 4; \\ -2 \mod p & \textp\equiv 3\bmod 4. \end W. Zhang and Z. Xu proved that, for any prime p \equiv 1 \pmod and integer \alpha\geq 1 , we have : E_\not \equiv 0 \pmod where \phi(n) is the
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
.


Asymptotic approximation

The Euler numbers grow quite rapidly for large indices as they have the following lower bound : , E_, > 8 \sqrt \left(\frac\right)^.


Euler zigzag numbers

The
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
of \sec x + \tan x = \tan\left(\frac\pi4 + \frac x2\right) is :\sum_^ \fracx^n, where is the Euler zigzag numbers, beginning with :1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, ... For all even , :A_n = (-1)^\frac E_n, where is the Euler number; and for all odd , :A_n = (-1)^\frac\frac, where is the
Bernoulli number In mathematics, the Bernoulli numbers 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, ...
. For every ''n'', :\frac\sin+\sum_^\frac\sin=\frac.


See also

*
Bell number In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy ...
*
Bernoulli number In mathematics, the Bernoulli numbers 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, ...
*
Dirichlet beta function In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of per ...
*
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...


References


External links

* * {{DEFAULTSORT:Euler Number Integer sequences Leonhard Euler