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Euler Number

The Euler numbers, also called the Secant Numbers or Zig Numbers, are defined for $\vert x\vert<\pi/2$ by

\mathop{\rm sech}\nolimits x-1 \equiv - {E_1^*x^2\over 2!} + {E_2^*x^4\over 4!} - {E_3^*x^6\over 6!} + \ldots
\end{displaymath} (1)

\sec x-1 \equiv {E_1^*x^2\over 2!} + {E_2^*x^4\over 4!} + {E_3^*x^6\over 6!} + \ldots,
\end{displaymath} (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

$\displaystyle E_1^*$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle E_2^*$ $\textstyle =$ $\displaystyle 5$  
$\displaystyle E_3^*$ $\textstyle =$ $\displaystyle 61$  
$\displaystyle E_4^*$ $\textstyle =$ $\displaystyle 1{,}385$  
$\displaystyle E_5^*$ $\textstyle =$ $\displaystyle 50{,}521$  
$\displaystyle E_6^*$ $\textstyle =$ $\displaystyle 2{,}702{,}765$  
$\displaystyle E_7^*$ $\textstyle =$ $\displaystyle 199{,}360{,}981$  
$\displaystyle E_8^*$ $\textstyle =$ $\displaystyle 19{,}391{,}512{,}145$  
$\displaystyle E_9^*$ $\textstyle =$ $\displaystyle 2{,}404{,}879{,}675{,}441$  
$\displaystyle E_{10}^*$ $\textstyle =$ $\displaystyle 370{,}371{,}188{,}237{,}525$  
$\displaystyle E_{11}^*$ $\textstyle =$ $\displaystyle 69{,}348{,}874{,}393{,}137{,}901$  
$\displaystyle E_{12}^*$ $\textstyle =$ $\displaystyle 15{,}514{,}534{,}163{,}557{,}086{,}905$  

(Sloane's A000364). The first few Prime Euler numbers $E_n$ occur for $n=2$, 3, 19, 227, 255, ... (Sloane's A014547) up to a search limit of $n=1415$.

The slightly different convention defined by

$\displaystyle E_{2n}$ $\textstyle =$ $\displaystyle (-1)^nE_n^*$ (3)
$\displaystyle E_{2n+1}$ $\textstyle =$ $\displaystyle 0$ (4)

is frequently used. These are, for example, the Euler numbers computed by the Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) function EulerE[n]. This definition has the particularly simple series definition
\mathop{\rm sech}\nolimits x-1 \equiv \sum_{k=0}^\infty {E_k x^k\over k!}
\end{displaymath} (5)

and is equivalent to
E_n=2^nE_n({\textstyle{1\over 2}}),
\end{displaymath} (6)

where $E_n(x)$ 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


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.'' 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, $E_n$.'' Ch. 5 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 39-42, 1987.

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© 1996-9 Eric W. Weisstein