he Euler numbers, also called the secant numbers or zig numbers, are defined for 
by
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(1)
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(2)
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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
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(3)
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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).
Some values of the (secant) Euler numbers are
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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(Sloane's A000364).
The slightly different convention defined by
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(16)
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(17)
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is frequently used. These are, for example, the Euler numbers computed by the Mathematica function EulerE[n]. This definition has the particularly simple series definition
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(18)
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and is equivalent to
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(19)
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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, ... (Sloane's A047893). The number of decimal digits in 
for 
, 1, ... are 1, 5, 139, 2372, 33699, ... (Sloane's A103235).
The first few prime Euler numbers 
occur for 
, 6, 38, 454, 510, ... (Sloane's A103234) up to a search limit of 
(Weisstein, Mar. 21, 2009). These correspond to 5, 61, 23489580527043108252017828576198947741, ... (Sloane's A092823). 
was proven to be prime by D. Broadhurst in 2002.
The Euler numbers have the asymptotic series
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(20)
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A more efficient asymptotic series is given by
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(21)
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(P. Luschny, pers. comm., 2007).
Expanding 
for even 
gives the identity
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(22)
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where the coefficient 
is interpreted as 
(Ely 1882; Fort 1948; Trott 2004, p. 69) and 
is a tangent number.
Stern (1875) showed that
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(23)
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iff 
. This result had been previously stated by Sylvester in 1861, but without proof.
Shanks (1968) defines a generalization of the Euler numbers by
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(24)
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Here,
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(25)
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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, ....
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Sloane |  |
| 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, ... |
SEE ALSO:Bernoulli Number, Euler Characteristic, Eulerian Number, Euler Polynomial, Euler Zigzag Number, Genocchi Number, Integer Sequence Primes, Lefschetz Number, Tangent Number
RELATED WOLFRAM SITES: http://functions.wolfram.com/IntegerFunctions/EulerE/
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. Amer. J. Math. 5, 337, 1882.
Fort, T. Finite Differences and Difference equation s 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. Crelle's J. 79, 67-98, 1861.
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.
