The zig numbers (also called the Euler numbers or the secant numbers) are defined using the secant or hyperbolic secant (the nth zig number is denoted E_{n}):

/ 2 \ / 4 \ / 6 \
| E x | | E x | | E x |
sech(x) = 1-| 1 | + | 2 | - | 3 | + ...
| ------ | | ------ | | ------ |
\ 2! / \ 4! / \ 6! /
-or-
/ 2 \ / 4 \ / 6 \
| E x | | E x | | E x |
sec(x) = 1+| 1 | + | 2 | + | 3 | + ...
| ------ | | ------ | | ------ |
\ 2! / \ 4! / \ 6! /

Where |x|=abs(x)<pi/2.

The zig numbers, combined with the zag numbers (associated with the tangent) form the set of alternating permutations. The first several zig numbers are the following:

`
E`_{0}=1

E_{1}=1

E_{2}=5

E_{3}=61

E_{4}=1385

E_{5}=50521

Aside from their use in combinatorics, the zig numbers can also be used to define the Euler irregular primes.

*Info gathered from *__http://mathworld.wolfram.com/EulerNumber.html__ and the prime numbers mailing list, primenumbers@yahoogroups.com.