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BellB

BellB[n]
gives the Bell number B_n.
BellB[n, x]
gives the Bell polynomial B_n(x).
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Bell polynomials satisfy the generating function relation .
  • The Bell numbers are given by B_n=B_n(1).
  • For certain special arguments, BellB automatically evaluates to exact values.
  • BellB can be evaluated to arbitrary numerical precision.
  • BellB automatically threads over lists.
EXAMPLESCLOSE ALL
Basic Examples   (2)
The tenth Bell number:
The fifth Bell polynomial:
The tenth Bell number:
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The fifth Bell polynomial:
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Scope   (4)
Evaluate numerically:
The precision of the output tracks the precision of the input:
BellB threads element-wise over lists:
TraditionalForm formatting:
Generalizations & Extensions   (1)
BellB can be applied to a power series:
Applications   (2)
BellB numbers versus their asymptotics:
Compute the first ten complementary Bell numbers:
Properties & Relations   (4)
Generating function:
Sum can give results involving BellB:
Use FullSimplify to simplify expressions involving BellB:
Compute Bell numbers directly from set partitions:
Possible Issues   (1)
The first argument of BellB must be a non-negative integer:
Neat Examples   (1)
Integral representation for Bell numbers by Cesàro:
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