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BellB[n]

gives the Bell number TemplateBox[{n}, BellB].

BellB[n,x]

gives the Bell polynomial TemplateBox[{n, x}, BellB2].

Details
Details and Options Details and Options
Background & Context
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
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BellB

BellB[n]

gives the Bell number TemplateBox[{n}, BellB].

BellB[n,x]

gives the Bell polynomial TemplateBox[{n, x}, BellB2].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Bell polynomials satisfy the generating function relation e^((e^t-1)x)=sum_(n=0)^(infty)(TemplateBox[{n, x}, BellB2]t^n)/(n!).
  • The Bell numbers are given by TemplateBox[{n}, BellB]=TemplateBox[{n, 1}, BellB2].
  • 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 ^(th) Bell polynomial and BellB[n] returns the ^(th) Bell number TemplateBox[{n}, BellB]=TemplateBox[{n, 1}, BellB2]. Bell polynomials can be determined from the exponential generating function . The Bell numbers also satisfy the recurrence relation B_(n+1)=sum_(k=0)^nTemplateBox[{n, k}, Binomial]B_k. 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 TemplateBox[{n, x}, BellB2]=sum_(k=0)^nY_(n,k)(x,...,x) and . Letting TemplateBox[{n, k}, StirlingS2] denote the Stirling number of the second kind, returned by StirlingS2, B_n=B_n(1)=sum_(k=0)^nTemplateBox[{n, k}, StirlingS2].

Examples

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Basic Examples  (2)

The tenth Bell number:

Wolfram Language code: BellB[10]

The fifth Bell polynomial:

Wolfram Language code: BellB[5, x]

Scope  (5)

Evaluate numerically:

Wolfram Language code: BellB[1000, 0.5]

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

Wolfram Language code: BellB[1000, 0.500000000000000000000000]

BellB threads element-wise over lists:

Wolfram Language code: BellB[{1, 2, 3, 4, 5, 6}]

BellB can be applied to a power series:

Wolfram Language code: BellB[n, Sin[x] + O[x] ^ 3]

TraditionalForm formatting:

Wolfram Language code: BellB[n]//TraditionalForm
Wolfram Language code: BellB[n, x]//TraditionalForm

Applications  (4)

BellB numbers versus their asymptotics:

Wolfram Language code: Show[ListPlot[Table[(Log[BellB[n]]/n), {n, 50}]], Plot[Log[n] + (Log[Log[n]] + 1) ((1/Log[n]) - 1) + (1/2) ((Log[Log[n]]/Log[n]))^2, {n, 5, 50}]]

Compute the first 10 complementary Bell numbers:

Wolfram Language code: Table[BellB[n, -1], {n, 10}]

Compare with an expression in terms of the Stirling number of the second kind:

Wolfram Language code: Table[Underoverscript[∑, k = 0, n](-1)^kStirlingS2[n, k], {n, 10}]

Verify an expression for the Bell number in terms of a Hessenberg determinant for the first few cases:

Wolfram Language code: Table[BellB[n + 1] == Det[SparseArray[{Band[{2, 1}] -> 1, Band[{1, 1}] -> 2, {j_, k_} /; j < k :> (-1)^j + kBinomial[k - 1, j - 1]}, {n, n}]], {n, 2, 9}]

The Bell numbers BellB[n] can be characterized as the unique set of numbers such that two certain Hankel determinants made from these numbers are both equal to BarnesG[n+2]. Verify for the first few cases:

Wolfram Language code: Table[BarnesG[n + 2] == Det[HankelMatrix[BellB[Range[0, n]], BellB[Range[n, 2n]]]] == Det[HankelMatrix[BellB[Range[n + 1]], BellB[Range[n + 1, 2n + 1]]]], {n, 9}]

Properties & Relations  (7)

The exponential generating function for BellB:

Wolfram Language code: ExponentialGeneratingFunction[BellB[n], n, x]

Compare with the explicit summation formula:

Wolfram Language code: Sum[BellB[n]x^n / n!, {n, 0, Infinity}]

Sum can give results involving BellB:

Wolfram Language code: Sum[k^n / k!, {k, 0, Infinity}, Assumptions -> n∈Integers && n ≥ 0]
Wolfram Language code: Sum[StirlingS2[n, k], {k, 0, n}]

The ^(th) moment of a PoissonDistribution is given by the ^(th) Bell polynomial in its mean :

Wolfram Language code: Moment[PoissonDistribution[μ], n]

Use FullSimplify to simplify expressions involving BellB:

Wolfram Language code: FullSimplify[Mod[BellB[k + p] + BellB[k] - BellB[k + 1], p], p ∈ Primes]

Compute Bell numbers directly from set partitions :

Wolfram Language code: Table[Sum[(n!/Underoverscript[∏, k = 1, n]k!^Subscript[m, k]Subscript[m, k]!)Boole[Subsuperscript[∑, k = 0, n]k Subscript[m, k] == n∧Subsuperscript[∑, k = 1, n]Subscript[m, k] == Subscript[m, 0]], Evaluate[Array[{Subscript[m, #], 0, n}&, n + 1, 0, Sequence]]], {n, 1, 5}]

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

Wolfram Language code: Table[Sum[(Multinomial@@p) / Product[k!, {k, Length /@ Split[p]}], {p, IntegerPartitions[n]}], {n, 10}]

Compare with the result of BellB:

Wolfram Language code: Table[BellB[k], {k, 1, 10}]

Compute Bell numbers using generalized Bell polynomials:

Wolfram Language code: Table[BellY[ConstantArray[1, {n, 2}]], {n, 10}]
Wolfram Language code: Table[BellB[n], {n, 10}]

Compute Bell polynomials using generalized Bell polynomials:

Wolfram Language code: Table[BellY[ConstantArray[{1, x}, n]], {n, 5}]
Wolfram Language code: Table[BellB[n, x], {n, 5}]

FindSequenceFunction can recognize the BellB sequence:

Wolfram Language code: Table[BellB[n], {n, 10}]
Wolfram Language code: FindSequenceFunction[%, n]

Possible Issues  (1)

The first argument of BellB must be a non-negative integer:

Wolfram Language code: BellB[1 / 2, x]

Neat Examples  (1)

Integral representation for Bell numbers by Cesàro:

Wolfram Language code: Im[Table[(2/π E)n! NIntegrate[E^E^E^I t Sin[n t], {t, 0, Pi}], {n, 10}]]
Wolfram Language code: BellB[Range[10]]
Wolfram Research (2007), BellB, Wolfram Language function, https://reference.wolfram.com/language/ref/BellB.html.

Text

Wolfram Research (2007), BellB, Wolfram Language function, https://reference.wolfram.com/language/ref/BellB.html.

CMS

Wolfram Language. 2007. "BellB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BellB.html.

APA

Wolfram Language. (2007). BellB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BellB.html

BibTeX

@misc{reference.wolfram_2026_bellb, author="Wolfram Research", title="{BellB}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/BellB.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_bellb, organization={Wolfram Research}, title={BellB}, year={2007}, url={https://reference.wolfram.com/language/ref/BellB.html}, note=[Accessed: 12-June-2026]}