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*Euler numbers ^{}* $E_{n}$ have the generating function $\displaystyle\frac{1}{\cosh{x}}$ such that

$\frac{1}{\cosh{x}}\;=:\;\sum_{{n=0}}^{\infty}\frac{E_{n}}{n!}\,x^{n}.$ |

They are integers but have no simple expression for calculating them. Their only regularities are that the numbers with odd index are all 0 and that

$\mbox{sgn}(E_{{2m}})\;=\;(-1)^{m}\qquad\mbox{for}\quad m=0,\,1,\,2,\,\ldots$ |

The Euler number have intimate relation to the Bernoulli numbers^{}. The first Euler numbers with even index are

$E_{0}=1,\quad E_{2}=-1,\quad E_{4}=5,\quad E_{6}=-61,\quad E_{8}=1385,\quad E_% {{10}}=-50521.$ |

- •
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 $\cosh{ix}=\cos{x}$, the division $1:\cos{x}$ correspondingly gives only terms with plus sign, i.e. it shows the absolute values

^{}of the Euler numbers. - •
The Euler numbers may also be obtained by using the Euler polynomials $E_{n}(x)$:

$E_{n}\;=\;2^{n}E_{n}\!\!\left(\!\frac{1}{2}\!\right)$ - •
If the Euler numbers $E_{k}$ are denoted as symbolic powers $E^{k}$, then one may write the equation

$(E\!+\!1)^{n}+(E\!-\!1)^{n}\;=\;0,$ which can be used as a recurrence relation for computing the values of the even-indexed Euler numbers. Cf. the Leibniz rule

^{}for derivatives of product $fg$.

## Mathematics Subject Classification

11B68*no label found*

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