Números de Catalan

José A. Alonso, 13-marzo-2017

Los números de Catalan forman la sucesión cuyo término general es

Los primeros números de Catalan son

   1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786

Los números de Catalan satisfacen la siguiente relación de recurrencia:

Asintóticamente, los números de Catalan crecen como:

considerando que el cociente entre el n-ésimo número de Catalan y la expresión de la derecha tiende hacia 1 cuando n tiende a infinito.

Definir las funciones

   catalan :: [Integer]
   grafica :: Int -> Int -> IO ()

tales que

catalan es la lista de términos de la sucesión de Catalan. Por ejemplo,

     λ> take 12 catalan
     [1,1,2,5,14,42,132,429,1430,4862,16796,58786]
     λ> length (show (catalan !! 50000))
     30096

(grafica a b) dibuja los n-ésimos términos de la sucesión de Catalan, para n entre a y b, junto con los de la expresión de la derecha de

Por ejemplo, (grafica 5 10) dibuja

y (grafica 55 60) dibuja

Soluciones

import Graphics.Gnuplot.Simple
 
catalan :: [Integer]
catalan = scanl (\c n -> c*2*(2*n-1) `div` (n+1)) 1 [1..]
 
grafica :: Int -> Int -> IO ()
grafica a b = 
  plotLists [Key Nothing]
            [[(fromIntegral n, fromIntegral (catalan !! n)) | n <- [a..b]]
            ,[(n,4**n/(n**(3/2)*(sqrt pi))) | n <- [c..d]]]
  where c, d :: Double
        c = fromIntegral a
        d = fromIntegral b

Posted in Medio

Tagged div, fromIntegral, Gráficas, Orden superior, pi, plotList, scanl, sqrt

12 Comments

Juanjo Ortega (juaorture)

13-marzo-2017 at 13:53

Responder

 
import Graphics.Gnuplot.Simple
 
catalan :: [Integer]
catalan = aux 1 1
 
aux :: Integer -> Integer -> [Integer]
aux n x = x : aux m ((product [1..2*n])`div`(m*product [1..n] ^ 2))
    where m = n + 1
 
grafica :: Int -> Int -> IO ()
grafica a b = plotLists [Key Nothing] ([[(x, f (fI x)) | x <- [a..b]],[(x,g (fI x)) | x <- [a..b]]] :: [[(Int,Double)]])
        where f n = (product [1..2*n])/((n+1)*product [1..n] ^ 2)
              g n = 4**n / ((n ** (3/2))*sqrt(pi))
              fI = fromIntegral

ignareeva
13-marzo-2017 at 16:36
Responder
catalan :: [Integer] catalan = 1: map (floor) [numerosCatalan n | n <- [0..]] where numerosCatalan 0 = 1 numerosCatalan n = (2*((2*n)+1))/(n+2) * numerosCatalan (n-1)
catalan :: [Integer] catalan = 1: map (floor) [numerosCatalan n | n <- [0..]] where numerosCatalan 0 = 1 numerosCatalan n = (2*((2*n)+1))/(n+2) * numerosCatalan (n-1)
monlagare
13-marzo-2017 at 17:11
Responder
catalan :: [Integer] catalan = [div (product [1..2*n]) (product [1..n+1] * product [1..n]) | n <- [0..]]
catalan :: [Integer] catalan = [div (product [1..2*n]) (product [1..n+1] * product [1..n]) | n <- [0..]]

eliguivil

13-marzo-2017 at 17:39

Responder

import Graphics.Gnuplot.Simple
 
catalan :: [Integer]
catalan = [(fac (2*n))`div`(fac (n+1) * fac n) | n <- [0..]]
  where
    fac n = product [1..n]
 
grafica :: Int -> Int -> IO ()
grafica a b = do plotLists [] [xs,ys]
  where
    xs = zip [a'..b'] catalan'
    ys :: [(Double,Double)]
    ys = zip [a'..b'] asin
    asin :: [Double]
    asin = [(4**n)/(n**(3/2)*sqrt pi) | n <- [a'..b']]
    a' = fromIntegral a
    b' = fromIntegral b
    catalan' :: [Double]
    catalan' = [fromIntegral $ (fac (2*n))`div`(fac (n+1) * fac n) | n <- [a..b]]
      where
        fac n = product [1..n]

eliguivil

13-marzo-2017 at 17:39

Responder

import Graphics.Gnuplot.Simple (plotLists)
 
catalan :: [Integer]
catalan = [(fac (2*n))`div`(fac (n+1) * fac n) | n <- [0..]]
  where
    fac n = product [1..n]
 
grafica :: Int -> Int -> IO ()
grafica a b = do plotLists [] [xs,ys]
  where
    xs = zip [a'..b'] catalan'
    ys :: [(Double,Double)]
    ys = zip [a'..b'] asin
    asin :: [Double]
    asin = [(4**n)/(n**(3/2)*sqrt pi) | n <- [a'..b']]
    a' = fromIntegral a
    b' = fromIntegral b
    catalan' :: [Double]
    catalan' = [fromIntegral $ (fac (2*n))`div`(fac (n+1) * fac n) | n <- [a..b]]
      where
        fac n = product [1..n]

marmerzaf

13-marzo-2017 at 18:05

Responder

listaCatalan :: [Integer]
listaCatalan  = [factorial (2*n) `div`( factorial (n+1) *  factorial n)| n <- [0..]]
factorial n = product [1.. n]
 
 
catalan n =factorial (2*n)/( factorial (n+1) *  factorial n)
cn n = 4 ** n / (n**(3/2)* n**1/2)
 
grafica3 :: Int -> Int -> IO ()
grafica3 a b =  plotLists [] ([[(x1, catalan( fromIntegral x1)) | x1 <- [a..b]],[(y1, cn(fromIntegral y1))| y1 <- [a..b]]] :: [[(Int,Double)]])

antmorper3

13-marzo-2017 at 18:36

Responder

catalan :: [Integer]
catalan = [aux n | n <- [0..]]
 where aux 0 = 1
       aux n = div (2*aux (n-1)*(2*n-1)) (n+1)
 
grafica :: Int -> Int -> IO ()
grafica a b = do plotLists [Key Nothing] ([[(n,fromInteger(catalan!!n)) | n <- [a..b]]
                                           ,[(n,f (fromIntegral n)) | n <- [a..b]]] :: [[(Int,Double)]])
 where f n = 4**n / (n**(3/2)*sqrt (pi))

enrnarbej

13-marzo-2017 at 19:10

Responder

catalan :: [Integer]
catalan = map fst cata
        where
         cata = (1,0) : map ((c,n) -> (2*c*(2*n+1) `div` (n+2),n+1)) cata
 
-- La gráfica la representaremos en Maxima
-- catalan (n) := (2*n)! / ((n+1)!*n!)$
-- catalanAprox (n) := float ((4^n) / (n^(3/2)*sqrt (%pi)))$
-- grafica (a,b) := wxplot2d([catalan (x),catalanAprox (x)],[x,a,b])$

albcercid

13-marzo-2017 at 19:15

Responder

 
catalan :: [Integer]
catalan = aux 1 0
   where aux a b = a:aux (div (2*(2*b + 1)*a) (b + 2)) (b + 1)
 
grafica :: Int -> Int -> IO ()
grafica a b = plotFuncs [] [c..d] [graf2,catalan2]
       where c = fromIntegral a
             d = fromIntegral b
 
graf2 :: Double -> Double
graf2 x = (4**x)/(((x**(1.5))*sqrt (pi)))
 
 
catalan2 :: Double -> Double
catalan2 n =(fc (2*n))/(((fc (n))^2)*(n+1))
        where   fc x = product [1..x]

glovizcas

13-marzo-2017 at 19:26

Responder

catalan :: [Integer]
catalan =  [(fact(2*n))`div`(fact(n+1)*fact n) | n <- [0..]]
 
fact :: Integer -> Integer
fact 0 = 1
fact n = n*fact (n-1)
 
grafica4 :: Int -> Int -> IO ()
grafica4 a b = plotLists [Key Nothing]
    ([[(x, fromInteger( last(take x(catalan))) ) | x <- [a..b]], 
    [(x,(expresion (fromIntegral x)))| x <- [a..b]]] :: [[(Int,Double)]] )
 
expresion x = (4.0**x)/(x**(1.5)*pi**(0.5))

antdursan

14-marzo-2017 at 16:58

Responder

catalan :: [Integer]
catalan = [div (fact (2*n)) (((fact (n+1))*(fact n))) | n <- [0..]]
              where fact 0 = 1
                    fact n = n*(fact (n-1))
 
catalanAprox :: [Double]
catalanAprox = [(4.0**(fromIntegral x))/((fromIntegral x)**(1.5)*pi**(0.5)) | x <- [1..]]
 
grafica :: Int -> Int -> IO ()
grafica a b = plotLists [] [[(x,fromInteger (head (drop (x) catalan))) | x <- [a..b]], [(x, (head (drop (x-1) (catalanAprox)))) | x <- [a..b]]]

cescarde

15-marzo-2017 at 00:31

Responder

catalan :: [Integer]
catalan = [c n | n <- [0..]]
 
c :: Integer -> Integer
c 0 = 1
c n = f (2*n) `div` ((f (n+1))*f n)
  where f x = product [1..x]
 
 
grafica :: Int -> Int -> IO ()
grafica a b = do plotLists [h] ([[(n,f (catalan!!n)) | n <- [a..b]],
                                 [(n,g (f1 n)) | n <- [a..b]]] :: [[(Int,Double)]])
  where g n = 4**n / (n**(3/2)*sqrt (pi))
        h = Key Nothing
        f = fromInteger
        f1 = fromIntegral

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