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combinat

  

bell

  

compute Bell numbers

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

bell(n)

Parameters

n

-

expression

Description

• 

The procedure bell computes the nth 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:

ⅇⅇx1=n=0bellnxnn!

• 

The Bell numbers are computed using the umbral definition :

belln+1=bell+1n

  

where bell()^n represents bell(n).

• 

For example:

bell3=bell2+2bell1+1

bell4=bell3+3 bell2+3 bell1+1

belln+1=i=0nnibelli

if 0<n

• 

The nth 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

withcombinat&comma;bell

bell

(1)

bell1

1

(2)

bell4

15

(3)

bell1

1

(4)

belln

combinat:-belln

(5)

See Also

BellB

binomial

combinat

 


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