Encyclopedia > Euler numbers

  Article Content

Euler number

Redirected from Euler numbers

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Shinnecock Hills, New York

... age or older. The average household size is 2.45 and the average family size is 3.00. In the town the population is spread out with 13.8% under the age of 18, 34.0% from ...

 
 
 
This page was created in 21.1 ms