Editing Euler numbers (section)

{{Use American English|date = March 2019}}
{{short description|Integers occurring in the coefficients of the Taylor series of 1/cosh t}}
{{confused|Eulerian number|Euler's number}}
{{other uses|List of things named after Leonhard Euler#Numbers}}
In [[mathematics]], the '''Euler numbers''' are a [[sequence]] ''E<sub>n</sub>'' of [[integer]]s {{OEIS|A122045}} defined by the [[Taylor series]] expansion

:<math>\frac{1}{\cosh t} = \frac{2}{e^{t} + e^ {-t} } = \sum_{n=0}^\infty  \frac{E_n}{n!} \cdot t^n</math>,

where <math>\cosh (t)</math> is the [[Hyperbolic function|hyperbolic cosine function]]. The Euler numbers are related to a special value of the [[Euler polynomials]], namely:
:<math>E_n=2^nE_n(\tfrac 12).</math>

The Euler numbers appear in the [[Taylor series]] expansions of the [[Trigonometric functions|secant]] and [[hyperbolic secant]] functions. The latter is the function in the definition. They also occur in [[combinatorics]], specifically when counting the number of [[alternating permutation]]s of a set with an even number of elements.