Added Python file for Catalan Numbers (Cn) - A sequence of natural numbers that appear in various counting problems.

Author: Rishabh Bafna

maths/series/catalan_numbers.py

@@ -0,0 +1,77 @@
+"""
+Catalan Numbers
+Reference: https://en.wikipedia.org/wiki/Catalan_number
+
+Catalan numbers C_n are a sequence of natural numbers that occur in
+various counting problems, often recursively defined.
+They can be computed directly using binomial coefficients:
+C_n = 1/(n+1) * C(2n, n)
+"""
+
+from __future__ import annotations
+import math
+import doctest
+
+
+def catalan_numbers(count: int) -> list[int]:
+    """
+    Generates the first 'count' Catalan numbers (C_0, C_1, ..., C_{count-1}).
+
+    :param count: The number of Catalan numbers to generate.
+    :return: A list of integers representing the Catalan numbers.
+
+    Examples:
+    >>> catalan_numbers(0)
+    []
+    >>> catalan_numbers(1)
+    [1]
+    >>> catalan_numbers(5)
+    [1, 1, 2, 5, 14]
+    >>> # C_6 = 42, C_7 = 132
+    >>> catalan_numbers(8)
+    [1, 1, 2, 5, 14, 42, 132, 429]
+    >>> catalan_numbers(-5)
+    Traceback (most recent call last):
+        ...
+    ValueError: Count must be a non-negative integer.
+    """
+    if not isinstance(count, int) or count < 0:
+        raise ValueError("Count must be a non-negative integer.")
+
+    if count == 0:
+        return []
+
+    sequence: list[int] = []
+
+    for n in range(count):
+        # Calculate C_n using the formula: C_n = (1 / (n + 1)) * (2n choose n)
+        # Using math.comb for direct, efficient calculation of binomial coefficient
+        # math.comb(N, K) computes N! / (K! * (N - K)!)
+        try:
+            binomial_coeff = math.comb(2 * n, n)
+            c_n = binomial_coeff // (n + 1)
+            sequence.append(c_n)
+        except AttributeError:
+            # Fallback for older Python versions (< 3.8) without math.comb
+            # This uses the factorial-based formula: (2n)! / ((n+1)! n!)
+            if n == 0:
+                sequence.append(1)
+            else:
+                numerator = math.factorial(2 * n)
+                denominator = math.factorial(n + 1) * math.factorial(n)
+                sequence.append(numerator // denominator)
+
+    return sequence
+
+
+if __name__ == "__main__":
+    doctest.testmod()
+
+    try:
+        num = int(input("Enter the number of Catalan numbers to generate: "))
+        if num < 0:
+             print("Please enter a non-negative integer.")
+        else:
+            print(f"The first {num} Catalan numbers are: {catalan_numbers(num)}")
+    except ValueError:
+        print("Invalid input. Please enter an integer.")