Documentation

Infinite zero-based table of binomial coefficients (also known as Pascal triangle): binomial !! n !! k == n! / k! / (n - k)!.

>>> take 5 (map (take 5) binomial)
[[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1]]

Complexity: binomial !! n !! k is O(n) bits long, its computation takes O(k n) time and forces thunks binomial !! n !! i for 0 <= i <= k. Use the symmetry of Pascal triangle binomial !! n !! k == binomial !! n !! (n - k) to speed up computations.

One could also consider binomial to compute stand-alone values.

Infinite zero-based table of Stirling numbers of the first kind.

>>> take 5 (map (take 5) stirling1)
[[1],[0,1],[0,1,1],[0,2,3,1],[0,6,11,6,1]]

Complexity: stirling1 !! n !! k is O(n ln n) bits long, its computation takes O(k n^2 ln n) time and forces thunks stirling1 !! i !! j for 0 <= i <= n and max(0, k - n + i) <= j <= k.

One could also consider unsignedStirling1st to compute stand-alone values.

Infinite zero-based table of Stirling numbers of the second kind.

>>> take 5 (map (take 5) stirling2)
[[1],[0,1],[0,1,1],[0,1,3,1],[0,1,7,6,1]]

Complexity: stirling2 !! n !! k is O(n ln n) bits long, its computation takes O(k n^2 ln n) time and forces thunks stirling2 !! i !! j for 0 <= i <= n and max(0, k - n + i) <= j <= k.

One could also consider stirling2nd to compute stand-alone values.

Infinite one-based table of Lah numbers. lah !! n !! k equals to lah(n + 1, k + 1).

>>> take 5 (map (take 5) lah)
[[1],[2,1],[6,6,1],[24,36,12,1],[120,240,120,20,1]]

Complexity: lah !! n !! k is O(n ln n) bits long, its computation takes O(k n ln n) time and forces thunks lah !! n !! i for 0 <= i <= k.

Infinite zero-based table of Eulerian numbers of the first kind.

>>> take 5 (map (take 5) eulerian1)
[[],[1],[1,1],[1,4,1],[1,11,11,1]]

Complexity: eulerian1 !! n !! k is O(n ln n) bits long, its computation takes O(k n^2 ln n) time and forces thunks eulerian1 !! i !! j for 0 <= i <= n and max(0, k - n + i) <= j <= k.

Infinite zero-based table of Eulerian numbers of the second kind.

>>> take 5 (map (take 5) eulerian2)
[[],[1],[1,2],[1,8,6],[1,22,58,24]]

Complexity: eulerian2 !! n !! k is O(n ln n) bits long, its computation takes O(k n^2 ln n) time and forces thunks eulerian2 !! i !! j for 0 <= i <= n and max(0, k - n + i) <= j <= k.

Infinite zero-based sequence of Bernoulli numbers, computed via connection with stirling2.

>>> take 5 bernoulli
[1 % 1,(-1) % 2,1 % 6,0 % 1,(-1) % 30]

Complexity: bernoulli !! n is O(n ln n) bits long, its computation takes O(n^3 ln n) time and forces thunks stirling2 !! i !! j for 0 <= i <= n and 0 <= j <= i.

One could also consider bernoulli to compute stand-alone values.

The same sequence as euler', but with type [a] instead of [Ratio a] as the denominators in euler' are always 1.

>>> take 10 euler :: [Integer]
[1,0,-1,0,5,0,-61,0,1385,0]

Infinite zero-based list of the n-th order Euler polynomials evaluated at 1. The algorithm used was derived from Algorithms for Bernoulli numbers and Euler numbers by Kwang-Wu Chen, third formula of the Corollary in page 7. Element-by-element division of sequences A1986631 and A006519 in OEIS.

>>> take 10 eulerPolyAt1 :: [Rational]
[1 % 1,1 % 2,0 % 1,(-1) % 4,0 % 1,1 % 2,0 % 1,(-17) % 8,0 % 1,31 % 2]

Faulhaber's formula.

>>> sum (map (^ 10) [0..100])
959924142434241924250
>>> sum $ zipWith (*) (faulhaberPoly 10) (iterate (* 100) 1)
959924142434241924250 % 1