The secant numbers 
,
also called the zig numbers or the positive Euler
numbers 
,
are defined by the generating function given
by the Maclaurin series of 
, indexed by the nonnegative
integer 
.
They also count alternating permutations
on 
symbols, with a permutation
and its reverse counted as equivalent. Here 
, and the first few 
for 
, 2, ... are 1, 5, 61, 1385, ... (OEIS A000364).
For example, the reversal-nonequivalent alternating permutations on 2 and 4 symbols are 
, and 
, 
, 
, 
, 
, respectively.
The secant numbers have the generating function
 |  |  |
(1)
|
 |  |  |
(2)
|
for 
.
