Definition:Euler Numbers

Definition
The Euler Numbers $E_n$ are a sequence of integers defined by the exponential generating function:


 * $\displaystyle \operatorname{sech} x = \frac {2 e^x} {e^{2 x} + 1} = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$

where $|x| < \dfrac \pi 2 $.

The values of the first Euler numbers are:


 * $1, 0, -1, 0, 5, 0, -61, 0, 1385, \ldots$

Also known as
The Euler numbers are also known as the Secant numbers or Zig numbers.