The factorial function (A000142).

Faster implementation of the factorial function

Naive implementation of factorial

"Swing factorial" algorithm

Prime factorization of the factorial (using the "swing factorial" algorithm)

Prime factorizaiton of the "swing factorial" function)

The double factorial

Faster implementation of the double factorial function

Naive implementation of the double factorial (A006882).

Binomial numbers (A007318). Note: This is zero for n<0 or k<0; see also signedBinomial below.

Faster implementation of binomial

A007318. Note: This is zero for n<0 or k<0; see also signedBinomial below.

The extension of the binomial function to negative inputs. This should satisfy the following properties:

for n,k >=0 : signedBinomial n k == binomial n k
for any n,k : signedBinomial n k == signedBinomial n (n-k) 
for k >= 0  : signedBinomial (-n) k == (-1)^k * signedBinomial (n+k-1) k

Note: This is compatible with Mathematica's Binomial function.

A given row of the Pascal triangle; equivalent to a sequence of binomial numbers, but much more efficient. You can also left-fold over it.

pascalRow n == [ binomial n k | k<-[0..n] ]

Catalan numbers. OEIS:A000108.

Catalan's triangle. OEIS:A009766. Note:

catalanTriangle n n == catalan n
catalanTriangle n k == countStandardYoungTableaux (toPartition [n,k])

Rows of (signed) Stirling numbers of the first kind. OEIS:A008275. Coefficients of the polinomial (x-1)*(x-2)*...*(x-n+1). This function uses the recursion formula.

(Signed) Stirling numbers of the first kind. OEIS:A008275. This function uses signedStirling1stArray, so it shouldn't be used to compute many Stirling numbers.

Argument order: signedStirling1st n k

(Unsigned) Stirling numbers of the first kind. See signedStirling1st.

Stirling numbers of the second kind. OEIS:A008277. This function uses an explicit formula.

Argument order: stirling2nd n k

Bernoulli numbers. bernoulli 1 == -1%2 and bernoulli k == 0 for k>2 and odd. This function uses the formula involving Stirling numbers of the second kind. Numerators: A027641, denominators: A027642.

Bell numbers (Sloane's A000110) from B(0) up to B(n). B(0)=B(1)=1, B(2)=2, etc.

The Bell numbers count the number of set partitions of a set of size n

See http://en.wikipedia.org/wiki/Bell_number

The n-th Bell number B(n), using the Stirling numbers of the second kind. This may be slower than using bellNumbersArray.