# nLab Euler number

Several numbers are named after Euler.

# Contents

## Euler’s number

Euler’s number is the irrational number

$e := \sum_{n \in \mathbb{N}} \frac{1}{n!} \,.$

This is the groupoid cardinality of the groupoid $core(FinSet)$, the core of FinSet.

## Euler numbers

The Euler numbers $E_n$, $n\geq 0$ are a sequence of integers defined via the generating function

$\frac{1}{ch t}= \frac{2}{e^t+e^{-t}} = \sum_{n\geq 0} E_n \frac{t^n}{n!}$

All odd-numbered members of the sequence vanish: $E_{2k-1}=0$ for $k\in\mathbb{N}$. $E_0=1$, $E_2 = -1$, $E_4 = 5$, see Euler number at wikipedia for more and the Abramowitz–Stegun handbook for many Euler numbers.

Euler numbers generalize to Euler polynomials $E_n(x)$ defined via the generating function

$\frac{2e^{tx}}{e^t+1}=\sum_{n\geq 0} E_n(x) \frac{t^n}{n!},$

so that $E_n = 2^n E_n(\frac{1}{2})$. The Euler polynomials satisfy the recursion $E_n(x+1)+E_n(x)=2x^n$ and may be conversely expressed via Euler numbers as

$E_n(x) = \sum_k \left(\array{n\\ k}\right) \frac{E_k}{2^k}\left(x-\frac{1}{2}\right)^{n-k}$

There is also a complementarity formula $E_n(1-x)=(-1)^n E_n(x)$. See e.g. Louis Comtet, Advanced combinatorics, D. Reifel Publ. Co., Dordrecht-Holland, Boston 1974 djvu

## Eulerian numbers

There are also Eulerian numbers (forming a different, double sequence $A(n,k)$).

Revised on May 13, 2010 12:34:55 by Urs Schreiber (87.212.203.135)