Euler numbers
Euler numbers^{} ${E}_{n}$ have the generating function $\frac{1}{\mathrm{cosh}x}$ such that
$$\frac{1}{\mathrm{cosh}x}=:\sum _{n=0}^{\mathrm{\infty}}\frac{{E}_{n}}{n!}{x}^{n}.$$ 
They are integers but have no expression for calculating them. Their only are that the numbers with odd index (http://planetmath.org/IndexingSet) are all 0 and that
$$\text{sgn}({E}_{2m})={(1)}^{m}\mathit{\hspace{1em}\hspace{1em}}\text{for}\mathit{\hspace{1em}}m=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots}$$ 
The Euler number have intimate relation to the Bernoulli numbers^{}. The first Euler numbers with even index are
$${E}_{0}=1,{E}_{2}=1,{E}_{4}=5,{E}_{6}=61,{E}_{8}=1385,{E}_{10}=50521.$$ 

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One can by hand determine Euler numbers by performing the division of 1 by the Taylor series^{} of hyperbolic cosine^{} (cf. Taylor series via division and Taylor series of hyperbolic functions). Since $\mathrm{cosh}ix=\mathrm{cos}x$, the division $1:\mathrm{cos}x$ correspondingly gives only terms with plus sign, i.e. it shows the absolute values^{} of the Euler numbers.

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The Euler numbers may also be obtained by using the Euler polynomials^{} ${E}_{n}(x)$:
$${E}_{n}={\mathrm{\hspace{0.33em}2}}^{n}{E}_{n}\left(\frac{1}{2}\right)$$ 
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If the Euler numbers ${E}_{k}$ are denoted as symbolic powers ${E}^{k}$, then one may write the equation
$${(E+1)}^{n}+{(E1)}^{n}=\mathrm{\hspace{0.33em}0},$$ which can be used as a recurrence relation for computing the values of the evenindexed Euler numbers. Cf. the Leibniz rule^{} for derivatives of product $fg$.
Title  Euler numbers 

Canonical name  EulerNumbers 
Date of creation  20141202 17:43:40 
Last modified on  20141202 17:43:40 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  11 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 11B68 
Related topic  GudermannianFunction 
Related topic  BernoulliNumber 
Related topic  InverseGudermannianFunction 
Related topic  HermiteNumbers 