fix

Author: Rishabh Bafna

maths/series/catalan_numbers.py

@@ -4,8 +4,8 @@
 
 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)
+This implementation uses the dynamic programming approach based on the
+recursive definition: C_{n+1} = sum_{i=0}^{n} C_i C_{n-i}
 """
 
 from __future__ import annotations
@@ -32,44 +32,44 @@ def catalan_numbers(count: int) -> list[int]:
         ...
     ValueError: Count must be a non-negative integer.
     """
-    if not isinstance(count, int) or count < 0:
+    if not isinstance(count, int):
+        raise ValueError("Count must be a non-negative integer.")
+    if count < 0:
         raise ValueError("Count must be a non-negative integer.")
 
     if count == 0:
         return []
 
-    sequence: list[int] = []
+    # Initialize the sequence with the base case C_0 = 1
+    sequence: list[int] = [1]
+
+    # Dynamically compute C_n for n from 1 up to count - 1
+    # We already have C_0, so we need 'count - 1' more elements.
+    for n in range(1, count):
+        c_n = 0
+        # The recursive formula is C_n = sum_{i=0}^{n-1} C_i * C_{n-1-i}
+        # C_n is the sum of products of previously computed Catalan numbers.
+        # sequence currently holds [C_0, C_1, ..., C_{n-1}]
+        # The length of 'sequence' is 'n'
+        for i in range(n):
+            # C_i * C_{n-1-i}
+            c_n += sequence[i] * sequence[n - 1 - i]
 
-    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)
+        sequence.append(c_n)
 
     return sequence
 
 
 if __name__ == "__main__":
-    import math
     import doctest
+
     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.")
+        print("Invalid input. Please enter an integer.")