Números de Catalan

PorJosé A. Alonso 13-marzo-201726-marzo-2022

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

1	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 ()

1 2	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

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

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

12 Comentarios

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

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

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)

Responder

monlagare dice:

13-marzo-2017 a las 17:11

Haskell

catalan :: [Integer] catalan = [div (product [1..2*n]) (product [1..n+1] * product [1..n]) | n <- [0..]]

1
2

catalan :: [Integer]
catalan = [div (product [1..2*n]) (product [1..n+1] * product [1..n]) | n <- [0..]]

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]

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]

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]

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]

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)]])

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)]])

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))

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))

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])$

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])$

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]

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]

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))

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))

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]]]

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]]]

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

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

Responder

Números de Catalan

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