- Catalan
Catalan (n)

Get

`n`

th Catalan number.See Planetmath for more information.

- Combinations
Combinations (k,n)

Get all combinations of k numbers from 1 to n as a vector of vectors. (See also NextCombination)

- DoubleFactorial
DoubleFactorial (n)

Double factorial:

`n(n-2)(n-4)...`

See Planetmath for more information.

- Factorial
Factorial (n)

Factorial:

`n(n-1)(n-2)...`

See Planetmath for more information.

- FallingFactorial
FallingFactorial (n,k)

Falling factorial:

`(n)_k = n(n-1)...(n-(k-1))`

See Planetmath for more information.

- Fibonacci
Fibonacci (x)

Aliases:

`fib`

Calculate

`n`

th Fibonacci number. That is the number defined recursively byand`Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)`

.`Fibonacci(1) = Fibonacci(2) = 1`

See Wikipedia or Planetmath or Mathworld for more information.

- FrobeniusNumber
FrobeniusNumber (v,arg...)

Calculate the Frobenius number. That is calculate smallest number that cannot be given as a non-negative integer linear combination of a given vector of non-negative integers. The vector can be given as separate numbers or a single vector. All the numbers given should have GCD of 1.

See Mathworld for more information.

- GaloisMatrix
GaloisMatrix (combining_rule)

Galois matrix given a linear combining rule (a_1*x_1+...+a_n*x_n=x_(n+1)).

- GreedyAlgorithm
GreedyAlgorithm (n,v)

Find the vector

`c`

of non-negative integers such that taking the dot product with`v`

is equal to n. If not possible returns`null`

.`v`

should be given sorted in increasing order and should consist of non-negative integers.See Mathworld for more information.

- HarmonicNumber
HarmonicNumber (n,r)

Aliases:

`HarmonicH`

Harmonic Number, the

`n`

th harmonic number of order`r`

.- Hofstadter
Hofstadter (n)

Hofstadter's function q(n) defined by q(1)=1, q(2)=1, q(n)=q(n-q(n-1))+q(n-q(n-2)).

- LinearRecursiveSequence
LinearRecursiveSequence (seed_values,combining_rule,n)

Compute linear recursive sequence using Galois stepping.

- Multinomial
Multinomial (v,arg...)

Calculate multinomial coefficients. Takes a vector of

`k`

non-negative integers and computes the multinomial coefficient. This corresponds to the coefficient in the homogeneous polynomial in`k`

variables with the corresponding powers.The formula for

can be written as:`Multinomial(a,b,c)`

(a+b+c)! / (a!b!c!)

In other words, if we would have only two elements, then

is the same thing as`Multinomial(a,b)`

or`Binomial(a+b,a)`

.`Binomial(a+b,b)`

See Wikipedia, Planetmath, or Mathworld for more information.

- NextCombination
NextCombination (v,n)

Get combination that would come after v in call to combinations, first combination should be

. This function is useful if you have many combinations to go through and you don't want to waste memory to store them all.`[1:k]`

For example with Combinations you would normally write a loop like:

`for n in Combinations (4,6) do ( SomeFunction (n) );`

But with NextCombination you would write something like:

`n:=[1:4]; do ( SomeFunction (n) ) while not IsNull(n:=NextCombination(n,6));`

See also Combinations.

- Pascal
Pascal (i)

Get the Pascal's triangle as a matrix. This will return an

`i`

+1 by`i`

+1 lower diagonal matrix that is the Pascal's triangle after`i`

iterations.See Planetmath for more information.

- Permutations
Permutations (k,n)

Get all permutations of

`k`

numbers from 1 to`n`

as a vector of vectors.- RisingFactorial
RisingFactorial (n,k)

Aliases:

`Pochhammer`

(Pochhammer) Rising factorial: (n)_k = n(n+1)...(n+(k-1)).

See Planetmath for more information.

- StirlingNumberFirst
StirlingNumberFirst (n,m)

Aliases:

`StirlingS1`

Stirling number of the first kind.

See Planetmath or Mathworld for more information.

- StirlingNumberSecond
StirlingNumberSecond (n,m)

Aliases:

`StirlingS2`

Stirling number of the second kind.

See Planetmath or Mathworld for more information.

- Subfactorial
Subfactorial (n)

Subfactorial: n! times sum_{k=0}^n (-1)^k/k!.

- Triangular
Triangular (nth)

Calculate the

`n`

th triangular number.See Planetmath for more information.

- nCr
nCr (n,r)

Aliases:

`Binomial`

Calculate combinations, that is, the binomial coefficient.

`n`

can be any real number.See Planetmath for more information.

- nPr
nPr (n,r)

Calculate the number of permutations of size

`r`

of numbers from 1 to`n`

.