## Euler Number

The Euler numbers, also called the Secant Numbers or Zig Numbers, are defined for by

 (1)

 (2)

where sech is the Hyperbolic Secant and sec is the Secant. Euler numbers give the number of Odd Alternating Permutations and are related to Genocchi Numbers. The base e of the Natural Logarithm is sometimes known as Euler's number.

Some values of the Euler numbers are

(Sloane's A000364). The first few Prime Euler numbers occur for , 3, 19, 227, 255, ... (Sloane's A014547) up to a search limit of .

The slightly different convention defined by

 (3) (4)

is frequently used. These are, for example, the Euler numbers computed by the Mathematica (Wolfram Research, Champaign, IL) function EulerE[n]. This definition has the particularly simple series definition
 (5)

and is equivalent to
 (6)

where is an Euler Polynomial.

To confuse matters further, the Euler Characteristic is sometimes also called the Euler number.''

See also Bernoulli Number, Eulerian Number, Euler Polynomial, Euler Zigzag Number, Genocchi Number

References

Abramowitz, M. and Stegun, C. A. (Eds.). Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula.'' §23.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804-806, 1972.

Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 110-111, 1996.

Guy, R. K. Euler Numbers.'' §B45 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 101, 1994.

Knuth, D. E. and Buckholtz, T. J. Computation of Tangent, Euler, and Bernoulli Numbers.'' Math. Comput. 21, 663-688, 1967.

Sloane, N. J. A. Sequences A014547 and A000364/M4019 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Spanier, J. and Oldham, K. B. The Euler Numbers, .'' Ch. 5 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 39-42, 1987.