Merge pull request #221 from abhisheks008/main

Nth Catalan Numbers using Dynamic Programming

Author: prathimacode-hub

Dynamic Programming/Nth Catalan Numbers/Images/catalan-1.png

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+# Nth Catalan Numbers using Dynamic Programming
+Language used : **Python 3**
+
+## 🎯 Aim
+The aim of this script is to find out the n'th catalan numbers using Dynamic Programming.
+
+## 👉 Purpose
+The main purpose of this script is to show the implementation of Dynamic programming on finding out the n'th catalan numbers.
+
+## 📄 Description
+Catalan numbers are defined as a mathematical sequence that consists of positive integers, which can be used to find the number of possibilities of various combinations. 
+
+**Examples:**  
+```
+Input  : k = 12
+Output : 1 1 2 5 14 42 132 429 1430 4862 16796 58786 
+
+Input  : k = 10
+Output : 1 1 2 5 14 42 132 429 1430 4862 
+
+Input  : k = 8
+Output : 1 1 2 5 14 42 132 429 
+```
+
+## 📈 Workflow of the script
+- `catalan` - The main function of the script to find out the catalan numbers using Dynamic Approach.
+- `main` - This is the driver code for this code/script.
+- `k` - User given integer which signifies the upper range of the Catalan series.
+
+## 🧮 Algorithm
+Let's see how it works,
+- Create an array `catalan[]` for storing `ith` Catalan number.
+- Initialize, `catalan[0]` and `catalan[1]` = 1
+- Loop through `i = 2` to the given Catalan number `n`.
+- Loop throught `j = 0` to `j < i` and Keep adding value of `catalan[j] * catalan[i – j – 1]` into `catalan[i]`.
+- Finally, `return catalan[n]`.
+
+## 💻 Input and Output 
+- **Test Case 1 :**
+
+![](https://github.com/abhisheks008/PyAlgo-Tree/blob/main/Dynamic%20Programming/Nth%20Catalan%20Numbers/Images/catalan-1.png)
+
+- **Test Case 2 :**
+
+![](https://github.com/abhisheks008/PyAlgo-Tree/blob/main/Dynamic%20Programming/Nth%20Catalan%20Numbers/Images/catalan-2.png)
+
+- **Test Case 3 :**
+
+![](https://github.com/abhisheks008/PyAlgo-Tree/blob/main/Dynamic%20Programming/Nth%20Catalan%20Numbers/Images/catalan-3.png)
+
+
+## ⏰ Time and Space complexity
+- **Time Complexity:** `O(n^2)`. 
+- **Space Complexity:** `O(n)`.
+
+---------------------------------------------------------------
+## 🖋️ Author
+**Code contributed by, _Abhishek Sharma_, 2022 [@abhisheks008](github.com/abhisheks008)**
+
+[![forthebadge made-with-python](http://ForTheBadge.com/images/badges/made-with-python.svg)](https://www.python.org/)

Dynamic Programming/Nth Catalan Numbers/Images/catalan-2.png

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+# Problem Name: Nth Catalan Numbers
+# Approach: Dynamic Programming
+
+# -----------------------------------------------------------------------------------
+
+# Problem Statement: Catalan numbers are defined as a mathematical sequence 
+#                    that consists of positive integers, which can be used to 
+#                    find the number of possibilities of various combinations. 
+#                    Find out the catalan numbers in between the range from 0 to N.
+
+
+# -----------------------------------------------------------------------------------
+
+# Constraints:
+# n -> Integer taken as the range of the Catalan Numbers.
+# catalan[] -> Array of Catalan Numbers from 0 to n.
+
+# -----------------------------------------------------------------------------------
+
+# A dynamic programming based function to find nth
+# Catalan number
+ 
+def catalan(n):
+    if (n == 0 or n == 1):
+        return 1
+ 
+    # Table to store results of subproblems
+    catalan = [0]*(n+1)
+ 
+    # Initialize first two values in table
+    catalan[0] = 1
+    catalan[1] = 1
+ 
+    # Fill entries in catalan[]
+    # using recursive formula
+    for i in range(2, n + 1):
+        for j in range(i):
+            catalan[i] += catalan[j] * catalan[i-j-1]
+ 
+    # Return last entry
+    return catalan[n]
+ 
+ 
+# Driver Code
+print ("-- Nth Catalan Numbers using Dynamic Programming --")
+print ()
+print ("Enter the upper range : ")
+k = int(input())
+print ()
+print ("Printing Nth Catalan Numbers from 0 to {0}...".format(k))
+print ()
+print ("Here you go!")
+for i in range(k):
+	  print(catalan(i), end=" ")
+    
+    
+# -----------------------------------------------------------------------------------
+
+# Output:
+# -- Nth Catalan Numbers using Dynamic Programming --
+
+# Enter the upper range : 
+# 12
+
+# Printing Nth Catalan Numbers from 0 to 12...
+
+# Here you go!
+# 1 1 2 5 14 42 132 429 1430 4862 16796 58786 
+
+# -----------------------------------------------------------------------------------
+
+# Code Contributed by, Abhishek Sharma, 2022
+
+# -----------------------------------------------------------------------------------