View source for Euler number

{{Use American English|date = March 2019}}
{{Short description|Integer sequence}}
{{confused|Eulerian number|Euler's number}}
{{other uses|List of things named after Leonhard Euler#Euler's numbers}}
In [[mathematics]], the '''Euler numbers''' are a [[sequence]] ''E<sub>n</sub>'' of [[integer]]s {{OEIS|A122045}} defined by the [[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 {{math|cosh ''t''}} is the [[Hyperbolic function|hyperbolic cosine]]. The Euler numbers are related to a special value of the [[Euler polynomials]], namely:
:<math>E_n=2^nE_n(\tfrac 12).</math>

The Euler numbers appear in the [[Taylor series]] expansions of the [[Trigonometric functions|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.

== Examples ==
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> ||=||align=right| 1
|-
|''E''<sub>2</sub> ||=||align=right| −1
|-
|''E''<sub>4</sub> ||=||align=right| 5
|-
|''E''<sub>6</sub> ||=||align=right| −61
|-
|''E''<sub>8</sub> ||=||align=right| {{val|1385|fmt=gaps}}
|-
|''E''<sub>10</sub> ||=||align=right| {{val|−50521}}
|-
|''E''<sub>12</sub> ||=||align=right| {{val|2,702,765}}
|-
|''E''<sub>14</sub> ||=||align=right| {{val|−199,360,981}}
|-
|''E''<sub>16</sub> ||=||align=right| {{val|19,391,512,145}}
|-
|''E''<sub>18</sub> ||=||align=right| {{val|−2,404,879,675,441}}
|}
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==

===As an iterated sum===
An explicit formula for Euler numbers is:<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"] {{webarchive | url=https://web.archive.org/web/20120511083735/http://www.voofie.com/content/117/an-explicit-formula-for-the-euler-zigzag-numbers-updown-numbers-from-power-series/ | date= 2012-05-11 }}</ref>

:<math>E_{2n}=i\sum _{k=1}^{2n+1} \sum _{j=0}^k \binom{k}{j}\frac{(-1)^j(k-2j)^{2n+1}}{2^k i^k k}</math>

where {{mvar|i}} denotes the [[imaginary unit]] with {{math|''i''<sup>2</sup> {{=}} −1}}.

===As a sum over partitions===
The Euler number {{math|''E''<sub>2''n''</sub>}} can be expressed as a sum over the even [[Partition (number theory)|partitions]] of {{math|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 {{math|2''n'' − 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| class= math.NT | year= 2011 }}</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''<sub>1</sub> + ··· + ''k<sub>n</sub>''}} 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 {{mvar|k}}s to {{math|2''k''<sub>1</sub> + 4''k''<sub>2</sub> + ··· + 2''nk<sub>n</sub>'' {{=}} 2''n''}} and to {{math|''k''<sub>1</sub> + 3''k''<sub>2</sub> + ··· + (2''n'' − 1)''k<sub>n</sub>'' {{=}} 2''n'' − 1}}, 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!^2\,7!} + \frac{6}{1!\,3!\,5!}
	+\frac{1}{3!^3}- \frac{5}{1!^4\,5!} -\frac{10}{1!^3\,3!^2} + \frac{7}{1!^6\, 3!} - \frac{1}{1!^9}\right) \\
& = -50\,521.
\end{align}
</math>

===As a determinant===
{{math|''E''<sub>2''n''</sub>}} is 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>

===As an integral===
{{math|''E''<sub>2''n''</sub>}} is also given by the following integrals:
:<math>
\begin{align}
(-1)^{n}E_{2n}&=\int_{0}^{\infty} \frac{t^{2n}}{\cosh\frac{\pi t}2}\; dt =\left(\dfrac2\pi\right)^{2n+1}\int_{0}^{\infty} \frac{x^{2n}}{\cosh {x}}\; dx\\ \\ &=\left(\dfrac2\pi\right)^{2n}\int_0^1\log ^{2n}\left(\tan  \frac{\pi t}{4} \right)dt =\left(\dfrac2\pi\right)^{2n+1}\int_0^{\pi/2}\log ^{2n}\left(\tan  \frac{x}{2} \right)dx\\ \\ &= \dfrac{2^{2n+3}}{\pi^{2n+2}} \int_0^{\pi/2}x\log ^{2n}\left(\tan  x \right)\,dx = \left(\dfrac2\pi\right)^{2n+2} \int_0^{\pi }\frac{x}{2}\log ^{2n}\left(\tan  \frac{x}{2} \right)\,dx.\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>

==Euler zigzag numbers==
The [[Taylor series]] of {{math|sec ''x'' + tan ''x''}} is

:<math>\sum_{n=0}^{\infty} \frac{A_n}{n!}x^n,</math>

where {{mvar|A<sub>n</sub>}} is the [[Alternating permutation|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, ... {{OEIS|id=A000111}}

For all even {{mvar|n}},
:<math>A_n = (-1)^\frac{n}{2} E_n,</math>
where {{mvar|''E<sub>n</sub>''}} is the Euler number; and for all odd {{mvar|n}},
:<math>A_n = (-1)^\frac{n-1}{2}\frac{2^{n+1}\left(2^{n+1}-1\right)B_{n+1}}{n+1},</math>
where {{mvar|''B<sub>n</sub>''}} is the [[Bernoulli number]].

For every ''n'',
:<math>\frac{A_{n-1}}{(n-1)!}\sin{\left(\frac{n\pi}{2}\right)}+\sum_{m=0}^{n-1}\frac{A_m}{m!(n-m-1)!}\sin{\left(\frac{m\pi}{2}\right)}=\frac{1}{(n-1)!}.</math>{{cn|date=September 2016}}

==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}}

{{Classes of natural numbers}}

{{DEFAULTSORT:Euler Number}}
[[Category:Integer sequences]]
[[Category:Leonhard Euler]]