TOPICS
Search

Secant Number


The secant numbers S_k, also called the zig numbers or the Euler numbers E_n^*=|E_(2n)| numbers than can be defined either in terms of a generating function given as the Maclaurin series of secx or as the numbers of alternating permutations on n=2, 4, 6, ... symbols (where permutations that are the reverses of one another counted as equivalent). The first few S_n for n=1, 2, ... are 1, 5, 61, 1385, ... (OEIS A000364).

For example, the reversal-nonequivalent alternating permutations on n=2 and 4 numbers are {1,2}, and {1,3,2,4}, {1,4,2,3}, {2,1,4,3}, {2,3,1,4}, {2,4,1,3}, respectively.

The secant numbers have the generating function

secx=sum_(k=0)^(infty)(S_kx^(2k))/((2k)!)
(1)
=1+1/2x^2+5/(24)x^4+(61)/(720)x^6+....
(2)

See also

Alternating Permutation, Euler Number, Euler Zigzag Number, Secant, Tangent Number, Zig Number

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequence A000364/M4019 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Secant Number

Cite this as:

Weisstein, Eric W. "Secant Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SecantNumber.html

Subject classifications