# Definition:Euler Numbers

## Definition

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

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

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

### Recurrence Relation

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

### Sequence of Euler Numbers

The sequence of Euler numbers begins:

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

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

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

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

### Definition 2

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

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.