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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 En):

             /    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:

E0=1
E1=1
E2=5
E3=61
E4=1385
E5=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.

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