Euler number

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This page discusses a topic in number theory. For other quantities named "Euler number", see the list of topics named after Leonhard Euler.

In mathematics, in the area of number theory, the Euler numbers are a sequence En of integers defined by the following Taylor series expansion:

\frac{1}{\cosh t} = \frac{2}{\exp (t) + \exp (-t) } = \sum_{n=0}^{\infin} \frac{E_n}{n!} \cdot t^n

where cosh t is the hyperbolic cosine. The Euler numbers appear as a special value of the Euler polynomials.

The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A000364 in OEIS) have alternating signs. Some values are:

E0 = 1
E2 = −1
E4 = 5
E6 = −61
E8 = 1,385
E10 = −50,521
E12 = 2,702,765
E14 = −199,360,981
E16 = 19,391,512,145
E18 = −2,404,879,675,441

Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics; see alternating permutation.

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