bell - Maple Help

combinat

 bell
 compute Bell numbers

 Calling Sequence bell(n)

Parameters

 n - expression

Description

 • The procedure bell computes the $n$th Bell number if the argument n is an integer; otherwise, it returns the unevaluated function call. For the BellB polynomials see BellB.
 • The Bell numbers are defined by the exponential generating function:

${ⅇ}^{{ⅇ}^{x}-1}=\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{bell}\left(n\right){x}^{n}}{n!}$

 • The Bell numbers are computed using the umbral definition :

$\mathrm{bell}\left(n+1\right)={\left(\mathrm{bell}\left(\right)+1\right)}^{n}$

 where bell()^n represents bell(n).
 • For example:

 $\mathrm{bell}\left(3\right)=\mathrm{bell}\left(2\right)+2\mathrm{bell}\left(1\right)+1$ $\mathrm{bell}\left(n+1\right)=\sum _{i=0}^{n}\left(\genfrac{}{}{0}{}{n}{i}\right)\mathrm{bell}\left(i\right)$

if $0

 • The $n$th Bell number has several interesting interpretations, including

 the number of rhyming schemes in a stanza of $n$ lines the number of ways n unlike objects can be placed in $n$ like boxes the number of ways a product of $n$ distinct primes may be factored

 • The command with(combinat,bell) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{combinat},\mathrm{bell}\right)$
 $\left[{\mathrm{bell}}\right]$ (1)
 > $\mathrm{bell}\left(1\right)$
 ${1}$ (2)
 > $\mathrm{bell}\left(4\right)$
 ${15}$ (3)
 > $\mathrm{bell}\left(-1\right)$
 ${1}$ (4)
 > $\mathrm{bell}\left(n\right)$
 ${\mathrm{bell}}{}\left({n}\right)$ (5)