Euler numbers

Euler numbers $E_n$ have the generating function ${\displaystyle \frac{1}{\cosh x}}$ such that

$$\frac{1}{\cosh x}=:\sum _{n=0}^{\mathrm{\infty }}\frac{E_n}{n!}{x}^n.$$

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

$$E_0=1,E_2=-1,E_4=5,E_6=-61,E_8=1385,E_{10}=-50521.$$

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

• The Euler numbers may also be obtained by using the Euler polynomials $E_n(x)$:

  $$E_n={\mathrm{\hspace{0.33em}2}}^n{E}_n\left(\frac{1}{2}\right)$$

• If the Euler numbers $E_k$ are denoted as symbolic powers $E^k$, then one may write the equation

  $${(E+1)}^n+{(E-1)}^n=\mathrm{\hspace{0.33em}0},$$

  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 $fg$.