The Euler numbers, also called the secant numbers or zig numbers, are defined for  by

(1)

(2)

where  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.

A different sort of Euler number, the Euler number of a finite complex , is defined by

(3)

This Euler number is a topological invariant.

To confuse matters further, the Euler characteristic is sometimes also called the "Euler number" and numbers produced by the prime-generating polynomial  are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47). In this work, primes generated by that polynomial are termed Euler primes, and prime Euler numbers are terms Euler number primes.

Some values of the (secant) Euler numbers are

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)

(OEIS A000364).

The slightly different convention defined by

			(16)
			(17)

is frequently used. These are, for example, the Euler numbers computed by the Wolfram Language function EulerE[n]. This definition has the particularly simple series definition

(18)

and is equivalent to

(19)

where  is an Euler polynomial.

The number of decimal digits in  for , 2, 4, ... are 1, 1, 1, 2, 4, 5, 7, 9, 11, 13, 15, 17, ... (OEIS A047893). The number of decimal digits in  for , 1, ... are 1, 5, 139, 2372, 33699, ... (OEIS A103235).

The Euler numbers have the asymptotic series

(20)

A more efficient asymptotic series is given by

(21)

(P. Luschny, pers. comm., 2007).

Expanding  for even  gives the identity

(22)

where the coefficient  is interpreted as  (Ely 1882; Fort 1948; Trott 2004, p. 69) and  is a tangent number.

Stern (1875) showed that

(23)

iff . This result had been previously stated by Sylvester in 1861, but without proof.

Shanks (1968) defines a generalization of the Euler numbers by

(24)

Here,

(25)

and  is  times the coefficient of  in the series expansion of . A similar expression holds for , but strangely not for  with . The following table gives the first few values of  for , 1, ....

	OEIS
1	A000364	1, 1, 5, 61, ...
2	A000281	1, 3, 57, 2763, ...
3	A000436	1, 8, 352, 38528, ...
4	A000490	1, 16, 1280, 249856, ...
5	A000187	2, 30, 3522, 1066590, ...
6	A000192	2, 46, 7970, 3487246, ...
7	A064068	1, 64, 15872, 9493504, ...
8	A064069	2, 96, 29184, 22634496, ...
9	A064070	2, 126, 49410, 48649086, ...
10	A064071	2, 158, 79042, 96448478, ...

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Caldwell, C. "The Top 20: Euler Irregular." http://primes.utm.edu/top20/page.php?id=25.

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

Ely, G. S. "Some Notes on the Numbers of Bernoulli and Euler." Amer. J. Math. 5, 337-341, 1882.

Fort, T. Finite Differences and Difference Equations in the Real Domain. Oxford, England: Clarendon Press, 1948.

Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, p. 47, 2000.

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

Hauss, M. Verallgemeinerte Stirling, Bernoulli und Euler Zahlen, deren Anwendungen und schnell konvergente Reihen für Zeta Funktionen. Aachen, Germany: Verlag Shaker, 1995.

Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent, Euler, and Bernoulli Numbers." Math. Comput. 21, 663-688, 1967.

Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub.,p. 124, 1993.

Shanks, D. "Generalized Euler and Class Numbers." Math. Comput. 21, 689-694, 1967.

Shanks, D. Corrigendum to "Generalized Euler and Class Numbers." Math. Comput. 22, 699, 1968.

Sloane, N. J. A. Sequences A000364/M4019, A014547, A047893, A092823, A103234, and A103235 in "The On-Line Encyclopedia of Integer Sequences."

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

Stern, M. A. "Zur Theorie der Euler Schen Zahlen." J. reine angew. Math. 79, 67-98, 1875.

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Young, P. T. "Congruences for Bernoulli, Euler, and Stirling Numbers." J. Number Th. 78, 204-227, 1999.