Add catalan_numbers.py (TheAlgorithms#4455)

Reviewed by @mrmaxguns. This is an implementation of Catalan Numbers.

dynamic_programming/catalan_numbers.py

@@ -0,0 +1,79 @@
+"""
+Print all the Catalan numbers from 0 to n, n being the user input.
+
+ * The Catalan numbers are a sequence of positive integers that
+ * appear in many counting problems in combinatorics [1]. Such
+ * problems include counting [2]:
+ * - The number of Dyck words of length 2n
+ * - The number well-formed expressions with n pairs of parentheses
+ *   (e.g., `()()` is valid but `())(` is not)
+ * - The number of different ways n + 1 factors can be completely
+ *   parenthesized (e.g., for n = 2, C(n) = 2 and (ab)c and a(bc)
+ *   are the two valid ways to parenthesize.
+ * - The number of full binary trees with n + 1 leaves
+
+ * A Catalan number satisfies the following recurrence relation
+ * which we will use in this algorithm [1].
+ * C(0) = C(1) = 1
+ * C(n) = sum(C(i).C(n-i-1)), from i = 0 to n-1
+
+ * In addition, the n-th Catalan number can be calculated using
+ * the closed form formula below [1]:
+ * C(n) = (1 / (n + 1)) * (2n choose n)
+
+ * Sources:
+ *  [1] https://brilliant.org/wiki/catalan-numbers/
+ *  [2] https://en.wikipedia.org/wiki/Catalan_number
+"""
+
+
+def catalan_numbers(upper_limit: int) -> "list[int]":
+    """
+    Return a list of the Catalan number sequence from 0 through `upper_limit`.
+
+    >>> catalan_numbers(5)
+    [1, 1, 2, 5, 14, 42]
+    >>> catalan_numbers(2)
+    [1, 1, 2]
+    >>> catalan_numbers(-1)
+    Traceback (most recent call last):
+    ValueError: Limit for the Catalan sequence must be ≥ 0
+    """
+    if upper_limit < 0:
+        raise ValueError("Limit for the Catalan sequence must be ≥ 0")
+
+    catalan_list = [0] * (upper_limit + 1)
+
+    # Base case: C(0) = C(1) = 1
+    catalan_list[0] = 1
+    if upper_limit > 0:
+        catalan_list[1] = 1
+
+    # Recurrence relation: C(i) = sum(C(j).C(i-j-1)), from j = 0 to i
+    for i in range(2, upper_limit + 1):
+        for j in range(i):
+            catalan_list[i] += catalan_list[j] * catalan_list[i - j - 1]
+
+    return catalan_list
+
+
+if __name__ == "__main__":
+    print("\n********* Catalan Numbers Using Dynamic Programming ************\n")
+    print("\n*** Enter -1 at any time to quit ***")
+    print("\nEnter the upper limit (≥ 0) for the Catalan number sequence: ", end="")
+    try:
+        while True:
+            N = int(input().strip())
+            if N < 0:
+                print("\n********* Goodbye!! ************")
+                break
+            else:
+                print(f"The Catalan numbers from 0 through {N} are:")
+                print(catalan_numbers(N))
+                print("Try another upper limit for the sequence: ", end="")
+    except (NameError, ValueError):
+        print("\n********* Invalid input, goodbye! ************\n")
+
+    import doctest
+
+    doctest.testmod()