# 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

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## Basic Examples(2)

The tenth Bell number:

The fifth Bell polynomial:

## Scope(4)

Evaluate numerically:

The precision of the output tracks the precision of the input:

## Generalizations & Extensions(1)

BellB can be applied to a power series:

## Applications(2)

BellB numbers versus their asymptotics:

Compute the first 10 complementary Bell numbers:

## Properties & Relations(8)

Generating function:

Sum can give results involving BellB:

Moment of PoissonDistribution is given by BellB polynomial in its mean :

Use FullSimplify to simplify expressions involving BellB:

Compute Bell numbers directly from set partitions :

Use IntegerPartitions to directly sum over terms that satisfy the constraints on indices:

Compare to the built-in function values:

Compute Bell numbers using generalized Bell polynomials:

FindSequenceFunction can recognize the BellB sequence:

The exponential generating function for BellB:

## Possible Issues(1)

The first argument of BellB must be a non-negative integer: ## Neat Examples(1)

Integral representation for Bell numbers by Cesàro: