Add Catalan numbers implementation using combinatorial formula

math/catalan_numbers.cpp

@@ -0,0 +1,57 @@
+#include <cassert>
+#include <cstdint>
+#include <iostream>
+
+/**
+ * @brief Compute the nth Catalan number using a combinatorial approach
+ *
+ * Catalan Numbers:
+ * Sequence: 1, 1, 2, 5, 14, 42, ...
+ *
+ * Applications:
+ * - Count of valid parentheses expressions
+ * - Number of Binary Search Trees with n nodes
+ * - Polygon triangulation
+ *
+ * Formula:
+ * Cn = (2n)! / ((n+1)! * n!)
+ *
+ * This implementation uses an iterative relation:
+ * Cn = C(n-1) * 2*(2n-1)/(n+1)
+ *
+ * Time Complexity: O(n)
+ * Space Complexity: O(1)
+ *
+ * Note:
+ * Works reliably for n <= 35 (fits in uint64_t)
+ */
+
+uint64_t catalan(uint32_t n) {
+    uint64_t res = 1;  // C0 = 1
+
+    for (uint32_t i = 0; i < n; i++) {
+        res = res * 2 * (2 * i + 1) / (i + 2);
+    }
+
+    return res;
+}
+
+// ------------------ TESTS ------------------
+
+static void test() {
+    assert(catalan(0) == 1);
+    assert(catalan(1) == 1);
+    assert(catalan(2) == 2);
+    assert(catalan(3) == 5);
+    assert(catalan(4) == 14);
+    assert(catalan(5) == 42);
+    assert(catalan(10) == 16796);
+}
+
+// ------------------ MAIN ------------------
+
+int main() {
+    test();
+    std::cout << "All tests passed!\n";
+    return 0;
+}