WOLFRAM

Wolfram Language & System Documentation Center

Upgrading from:

DiscreteMath`CombinatorialFunctions`

CatalanNumber and Subfactorial have been added to the built-in Mathematica kernel.
CatalanNumber and Subfactorial can now be evaluated numerically for noninteger arguments.
Subfactorial now accepts complex arguments.

CatalanNumber

Syntax for CatalanNumber is the same as in DiscreteMath`CombinatorialFunctions` in Version 5.2:

Version 5.2 << DiscreteMath`CombinatorialFunctions`; Table[CatalanNumber[n], {n, 1, 10}]
Wolfram Language code: Table[CatalanNumber[n], {n, 1, 10}]

CatalanNumber can now be evaluated numerically for noninteger arguments:

Version 5.2 N[CatalanNumber[1/2]] returns unevaluated
Wolfram Language code: N[CatalanNumber[1 / 2]]

CatalanNumber can now be evaluated for complex arguments:

Version 5.2 CatalanNumber[0.5 I] returns unevaluated
Wolfram Language code: CatalanNumber[0.5 I]

Subfactorial

Syntax for Subfactorial is the same as in DiscreteMath`CombinatorialFunctions` in Version 5.2:

Version 5.2 << DiscreteMath`CombinatorialFunctions`; Table[Subfactorial[n], {n, 10}]
Wolfram Language code: Table[Subfactorial[n], {n, 10}]

Subfactorial can now be evaluated numerically for noninteger arguments:

Version 5.2 N[Subfactorial[1/2]] returns unevaluated
Wolfram Language code: N[Subfactorial[1 / 2]]

Subfactorial can now be evaluated for complex arguments:

Version 5.2 Subfactorial[1.5 + I] returns unevaluated
Wolfram Language code: Subfactorial[1.5 + I]

Hofstadter's function

Define Hofstadter's function:

Version 5.2 << DiscreteMath`CombinatorialFunctions`; Hofstadter[100]
Wolfram Language code: Hofstadter[1] = Hofstadter[2] = 1; Hofstadter[n_Integer ? Positive] := Hofstadter[n] = Block[ {$RecursionLimit = Infinity}, Hofstadter[n - Hofstadter[n - 1]] + Hofstadter[n - Hofstadter[n - 2]] ] Hofstadter[100]