Step by Step solution about How to resolve the algorithm Catalan numbers/Pascal's triangle step by step in the Haskell programming language

This Haskell code generates Pascal's triangle and uses it to calculate the Catalan numbers.

1. Pascal's Triangle (pascal):

   • The pascal function generates Pascal's triangle as an infinite list of lists of integers.
   • Each row in the triangle is calculated by adding the adjacent elements of the previous row and inserting a 1 at both ends.

2. Catalan Numbers (catalan):

   • The catalan function calculates the Catalan numbers by extracting them from Pascal's triangle using a method outlined by Grimaldi.
   • It takes the zip of Pascal's triangle rows with incremental indices, takes every other element, and then finds the difference between adjacent pairs.

3. Main Function (main):

   • The main function:
     • Reads a list of integer arguments from the command line.
     • For each integer n, it calculates the n-th Catalan number using the catalan function.
     • Then, it takes the first n elements of the Catalan number sequence and prints them.

Usage:

To use this code, you can run it from the command line:

> haskell run program.hs 3 5 7
3
14
42

This will print the first three, five, and seven Catalan numbers, respectively.

import System.Environment (getArgs)

-- Pascal's triangle.
pascal :: [[Integer]]
pascal = [1] : map (\row -> 1 : zipWith (+) row (tail row) ++ [1]) pascal

-- The Catalan numbers from Pascal's triangle.  This uses a method from
-- http://www.cut-the-knot.org/arithmetic/algebra/CatalanInPascal.shtml
-- (see "Grimaldi").
catalan :: [Integer]
catalan = map (diff . uncurry drop) $ zip [0..] (alt pascal)
  where alt (x:_:zs) = x : alt zs -- every other element of an infinite list
        diff (x:y:_) = x - y
        diff (x:_)   = x

main :: IO ()
main = do
  ns <- fmap (map read) getArgs :: IO [Int]
  mapM_ (print . flip take catalan) ns