Euler number

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

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

(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 (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.

The Euler polynomials are constructed with the Euler numbers.

Euler number

See also: Euler number, Bernoulli polynomials, Combinatorics, E (mathematical constant), Euler's number, Euler characteristic, Hyperbolic secant, Integer, On-Line Encyclopedia of Integer Sequences