# BellB

BellB[n]

gives the Bell number .

BellB[n,x]

gives the Bell polynomial .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• The Bell polynomials satisfy the generating function relation .
• The Bell numbers are given by .
• For certain special arguments, BellB automatically evaluates to exact values.
• BellB can be evaluated to arbitrary numerical precision.
• BellB automatically threads over lists.

# Background & Context

• BellB is a mathematical function that returns a Bell number or polynomial. In particular, BellB[n,x] returns the  Bell polynomial and BellB[n] returns the  Bell number . Bell polynomials can be determined from the exponential generating function . The Bell numbers also satisfy the recurrence relation . The first few Bell polynomials are , while the first few Bell numbers are .
• The Bell polynomial is also called an exponential polynomial or, more explicitly, the "complete exponential Bell polynomial" and is sometimes denoted . Bell polynomials are named after mathematician and math expositor Eric Temple Bell, who wrote about them in 1934.
• The polynomial has the interpretation that if there are partitions of into parts, then . Furthermore, if there are total partitions of , then . For example, the set having elements can be partitioned into parts ways , part way ( ), parts ways ( , and ), and parts way ( ), giving . Since there are five total ways to partition , .
• The Bell polynomial and number are a special case of the BellY function, with and . Letting denote the Stirling number of the second kind, returned by StirlingS2, .

# Examples

open all close all

## Basic Examples(2)

The tenth Bell number:

 In:= Out= The fifth Bell polynomial:

 In:= Out= ## Neat Examples(1)

Introduced in 2007
(6.0)