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{{other uses|List of things named after Leonhard Euler#Euler—numbers}}
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In [[number theory]], the '''Euler numbers''' are a [[sequence]] ''E<sub>n</sub>'' of [[integer]]s {{OEIS|A122045}} defined by the following [[Taylor series]] expansion:
 
:<math>\frac{1}{\cosh t} = \frac{2}{e^{t} + e^ {-t} } = \sum_{n=0}^\infty  \frac{E_n}{n!} \cdot t^n\!</math>
 
where cosh&nbsp;''t'' is the [[Hyperbolic function|hyperbolic cosine]]. The Euler numbers appear as a special value of the [[Euler polynomials]].
 
The odd-indexed Euler numbers are all [[0 (number)|zero]]. The even-indexed ones {{OEIS|id=A028296}} have alternating signs. Some values are:
:''E''<sub>0</sub> = 1
:''E''<sub>2</sub> = −1
:''E''<sub>4</sub> = 5
:''E''<sub>6</sub> = −61
:''E''<sub>8</sub> = 1,385
:''E''<sub>10</sub> = −50,521
:''E''<sub>12</sub> = 2,702,765
:''E''<sub>14</sub> = −199,360,981
:''E''<sub>16</sub> = 19,391,512,145
:''E''<sub>18</sub> = −2,404,879,675,441
 
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.
 
The Euler numbers appear in the [[Taylor series]] expansions of the [[trigonometric function|secant]] and [[hyperbolic secant]] functions.  The latter is the function in the definition. They also occur in [[combinatorics]], specifically when counting the number of [[alternating permutation]]s of a set with an even number of elements.
 
==Explicit formulas==
 
===Iterated sum===
An explicit formula for Euler numbers is given by:<ref>[http://www.voofie.com/content/117/an-explicit-formula-for-the-euler-zigzag-numbers-updown-numbers-from-power-series/ Ross Tang, "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series"]{{dead link|date=October 2013}}</ref>
 
:<math>E_{2n}=i\sum _{k=1}^{2n+1} \sum _{j=0}^k {k\choose j}\frac{(-1)^j(k-2j)^{2n+1}}{2^k i^k k}</math>
 
where ''i'' denotes the [[imaginary unit]] with ''i''<sup>2</sup>=&minus;1.
 
===Sum over partitions===
The Euler number ''E''<sub>2''n''</sub> can be expressed as a sum over the even [[Partition (number theory)|partitions]] of 2''n'',<ref>{{cite journal|first1=David C.|last1= Vella|title=Explicit Formulas for Bernoulli and Euler Numbers|journal=Integers|volume=8|issue=1|pages=A1|year=2008|url= http://www.integers-ejcnt.org/vol8.html}}</ref>
 
:<math>  E_{2n} = (2n)! \sum_{0 \leq k_1, \ldots, k_n \leq n}~  \left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right)
\delta_{n,\sum mk_m }  \left( \frac{-1~}{2!} \right)^{k_1} \left( \frac{-1~}{4!} \right)^{k_2}
\cdots \left( \frac{-1~}{(2n)!} \right)^{k_n} ,</math>
 
as well as a sum over the odd partitions of 2''n''&nbsp;&minus;&nbsp;1,<ref>{{cite arxiv|eprint=1103.1585|first1= J.|last1=Malenfant|title=Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers}}</ref>
 
:<math> E_{2n} =  (-1)^{n-1} (2n-1)! \sum_{0 \leq k_1, \ldots, k_n \leq 2n-1}
\left( \begin{array} {c} K \\ k_1, \ldots , k_n \end{array} \right)
\delta_{2n-1,\sum (2m-1)k_m }  \left( \frac{-1~}{1!} \right)^{k_1}  \left( \frac{1}{3!} \right)^{k_2}
  \cdots \left( \frac{(-1)^n}{(2n-1)!} \right)^{k_n}  , </math>
 
where in both cases <math> K =k_1 + \cdots + k_n</math> and
:<math> \left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right)
          \equiv \frac{ K!}{k_1! \cdots k_n!}</math>
is a [[multinomial coefficient]].  The [[Kronecker delta]]'s in the above formulas restrict the sums over the ''k'''s to <math> 2k_1 + 4k_2 + \cdots +2nk_n=2n</math> and to
<math> k_1 + 3k_2 + \cdots +(2n-1)k_n=2n-1</math>, respectively.
 
As an example,
:<math>
\begin{align}
E_{10} & = 10! \left( - \frac{1}{10!} + \frac{2}{2!8!} + \frac{2}{4!6!}
- \frac{3}{2!^2 6!}- \frac{3}{2!4!^2} +\frac{4}{2!^3 4!} - \frac{1}{2!^5}\right) \\
& = 9! \left( - \frac{1}{9!} + \frac{3}{1!^27!} + \frac{6}{1!3!5!}
+\frac{1}{3!^3}- \frac{5}{1!^45!} -\frac{10}{1!^33!^2} + \frac{7}{1!^6 3!} - \frac{1}{1!^9}\right) \\
& = -50,521.
\end{align}
</math>
 
===Determinant===
''E''<sub>2''n''</sub> is also given by the [[determinant]]
 
:<math>
\begin{align}
E_{2n} &=(-1)^n (2n)!~ \begin{vmatrix}  \frac{1}{2!}& 1 &~& ~&~\\
                                                            \frac{1}{4!}&  \frac{1}{2!} & 1 &~&~\\
                                                                \vdots & ~  &  \ddots~~ &\ddots~~ & ~\\
                                                              \frac{1}{(2n-2)!}& \frac{1}{(2n-4)!}& ~&\frac{1}{2!} &  1\\
                                                              \frac{1}{(2n)!}&\frac{1}{(2n-2)!}& \cdots &  \frac{1}{4!} & \frac{1}{2!}\end{vmatrix}.
 
\end{align}
</math>
 
==Asymptotic approximation==
 
The Euler numbers grow quite rapidly for large indices as
they have the following lower bound
 
: <math> |E_{2 n}| > 8 \sqrt { \frac{n}{\pi} } \left(\frac{4 n}{ \pi e}\right)^{2 n}. </math>
 
==See also==
* [[Bell number]]
* [[Bernoulli number]]
* [[Euler–Mascheroni constant]]
 
==References==
{{Reflist}}
 
==External links==
* {{springer|title=Euler numbers|id=p/e036540}}
* {{MathWorld|urlname=EulerNumber|title=Euler number}}
 
{{DEFAULTSORT:Euler Number}}
[[Category:Integer sequences]]

Latest revision as of 11:44, 2 January 2015

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