Catalan (n)
Get n
th Catalan number.
See Planetmath for more information.
Combinations (k,n)
Get all combinations of k numbers from 1 to n as a vector of vectors. (See also NextCombination)
See Wikipedia for more information.
DoubleFactorial (n)
Double factorial: n(n-2)(n-4)...
See Planetmath for more information.
Factorial (n)
Factorial: n(n-1)(n-2)...
See Planetmath for more information.
FallingFactorial (n,k)
Falling factorial: (n)_k = n(n-1)...(n-(k-1))
See Planetmath for more information.
Fibonacci (x)
Aliases: fib
Calculate n
th Fibonacci number. That
is the number defined recursively by
Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n-2)
and
Fibonacci(1) = Fibonacci(2) = 1
.
See Wikipedia or Planetmath or Mathworld for more information.
FrobeniusNumber (v,arg...)
Calculate the Frobenius number. That is calculate largest 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.
GaloisMatrix (combining_rule)
Galois matrix given a linear combining rule (a_1*x_1+...+a_n*x_n=x_(n+1)).
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.
HarmonicNumber (n,r)
Aliases: HarmonicH
Harmonic Number, the n
th harmonic number of order r
.
That is, it is the sum of 1/k^r
for k
from 1 to n. Equivalent to sum k = 1 to n do 1/k^r
.
See Wikipedia for more information.
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)).
See Wikipedia for more information. The sequence is A005185 in OEIS.
LinearRecursiveSequence (seed_values,combining_rule,n)
Compute linear recursive sequence using Galois stepping.
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 Multinomial(a,b,c)
can be written as:
(a+b+c)! / (a!b!c!)
In other words, if we would have only two elements, then
Multinomial(a,b)
is the same thing as
Binomial(a+b,a)
or
Binomial(a+b,b)
.
See Wikipedia, Planetmath, or Mathworld for more information.
NextCombination (v,n)
Get combination that would come after v in call to
combinations, first combination should be [1:k]
. This
function is useful if you have many combinations to go through and you don't
want to waste memory to store them all.
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.
See Wikipedia for more information.
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 (k,n)
Get all permutations of k
numbers from 1 to n
as a vector of vectors.
RisingFactorial (n,k)
Aliases: Pochhammer
(Pochhammer) Rising factorial: (n)_k = n(n+1)...(n+(k-1)).
See Planetmath for more information.
StirlingNumberFirst (n,m)
Aliases: StirlingS1
Stirling number of the first kind.
See Planetmath or Mathworld for more information.
StirlingNumberSecond (n,m)
Aliases: StirlingS2
Stirling number of the second kind.
See Planetmath or Mathworld for more information.
Subfactorial (n)
Subfactorial: n! times sum_{k=0}^n (-1)^k/k!.
Triangular (nth)
Calculate the n
th triangular number.
See Planetmath for more information.
nCr (n,r)
Aliases: Binomial
Calculate combinations, that is, the binomial coefficient.
n
can be any real number.
See Planetmath for more information.
nPr (n,r)
Calculate the number of permutations of size
r
of numbers from 1 to n
.