Euler numbers - Definition and Overview

The Euler numbers are a sequence E_n 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 (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 expansion of the secant trigonometric function and the hyperbolic secant (which is the function in the definition), and they also occur in combinatorics.

The Euler polynomials are constructed with the Euler numbers.