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 
.
