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Euler number

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The Euler numbers are a sequence En of integers defined by the following Taylor series expansion:

<math>\frac{2}{\exp (t) + \exp (-t) } = \sum_{n=0}^{\infin} \frac{E_n}{n!} \cdot t^n</math>

(Note that e, the base of the natural logarithm, is also occasionally called Euler's number, as is the Euler characteristic.)

The odd-indexed Euler numbers are all zero. The even-indexed ones 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 expansion of the secant trigonometric function, and they also occur in combinatorics.

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