# Definition:Euler Numbers

## Definition

The Euler Numbers $E_n$ are a sequence of integers defined by the exponential generating function:

$\ds \sech x = \frac {2 e^x} {e^{2 x} + 1} = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$

where $\size x < \dfrac \pi 2$.

### Recurrence Relation

$E_n = \begin {cases} 1 & : n = 0 \\ 0 & : n > 0, n = 2 r + 1 \\ \ds -\sum_{k \mathop = 0}^{n - 1} \binom {2 n} {2 k} E_{2 k} & : n > 0, n = 2 r \end {cases}$

### Sequence of Euler Numbers

The sequence of Euler numbers begins:

 $\ds E_0$ $=$ $\ds 1$ $\ds E_2$ $=$ $\ds -1$ $\ds E_4$ $=$ $\ds 5$ $\ds E_6$ $=$ $\ds -61$ $\ds E_8$ $=$ $\ds 1385$ $\ds E_{10}$ $=$ $\ds -50 \, 521$ $\ds E_{12}$ $=$ $\ds 2 \, 702 \, 765$ $\ds E_{14}$ $=$ $\ds -199 \, 360 \, 981$ $\ds E_{16}$ $=$ $\ds 19 \, 391 \, 512 \, 145$ $\ds E_{18}$ $=$ $\ds -2 \, 404 \, 879 \, 675 \, 441$ $\ds E_{20}$ $=$ $\ds 370 \, 371 \, 188 \, 237 \, 525$ $\ds E_{22}$ $=$ $\ds -69 \, 348 \, 874 \, 393 \, 137 \, 901$ $\ds E_{24}$ $=$ $\ds 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905$

Odd index Euler numbers are all $0$.

## Alternative Form

An alternative form of the Euler numbers can often be found.

Usually denoted with the symbol $E_n^*$, they are generally not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

### Definition 1

 $\ds \sech x$ $=$ $\ds 1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {E_n^* x^{2 n} } {\paren {2 n}!}$ $\ds$ $=$ $\ds 1 - \frac {E_1^* x^2} {2!} + \frac {E_2^* x^4} {4!} - \frac {E_3^* x^6} {6!} + \cdots$

where $\size x < \dfrac \pi 2$.

### Definition 2

 $\ds \sec x$ $=$ $\ds 1 + \sum_{n \mathop = 1}^\infty \frac {E^*_n x^{2 n} } {\paren {2 n}!}$ $\ds$ $=$ $\ds 1 + \frac {E_1^* x^2} {2!} + \frac {E_2^* x^4} {4!} + \frac {E_3^* x^6} {6!} + \cdots$

where $\size x < \dfrac \pi 2$.

## Also known as

The Euler numbers are also known as the secant numbers or zig numbers.

## Also see

• Results about Euler Numbers can be found here.

## Source of Name

This entry was named for Leonhard Paul Euler.