feat: Add memoized version of factorial (#2964)

* Create check_even_odd.cpp

Implementation to Check if a number is Even or Odd using Bitwise Operator

* Update check_even_odd.cpp

* Create factorial_top_down_dp.cpp

* Delete dynamic_programming/factorial_top_down_dp.cpp

Deleted the one file as there was 2 files in the commit

* Create factorial_top_down_dp.cpp

* Delete bit_manipulation/check_even_odd.cpp

* Update dynamic_programming/factorial_top_down_dp.cpp

Co-authored-by: realstealthninja <68815218+realstealthninja@users.noreply.github.com>

* Update dynamic_programming/factorial_top_down_dp.cpp

Co-authored-by: realstealthninja <68815218+realstealthninja@users.noreply.github.com>

* Update dynamic_programming/factorial_top_down_dp.cpp

Co-authored-by: realstealthninja <68815218+realstealthninja@users.noreply.github.com>

* Update factorial_top_down_dp.cpp

modified

* Update factorial_top_down_dp.cpp

added __uint128_t for handling  fixed-width integer types

* Create memoised_factorial.cpp

* Rename memoised_factorial.cpp to factorial_memoization.cpp

* Update factorial_memoization.cpp

* Delete dynamic_programming/factorial_top_down_dp.cpp

deleted the file from dp folder

* Update math/factorial_memoization.cpp

Co-authored-by: realstealthninja <68815218+realstealthninja@users.noreply.github.com>

* Update factorial_memoization.cpp

added cstdint header and switched to uint64 Thanks

* fix: wrap factorial functions under math namespace

* chore: add scope specifier in test cases

* doc: add documentation to headers

* doc: add defintion of memoisation as well as rewrite brief

---------

Co-authored-by: realstealthninja <68815218+realstealthninja@users.noreply.github.com>

math/factorial_memoization.cpp

@@ -0,0 +1,73 @@
+/**
+ * @file
+ * @brief [Factorial](https://en.wikipedia.org/wiki/Factorial) calculation using recursion and [memoization](https://en.wikipedia.org/wiki/Memoization) 
+ * @details
+ * This program computes the factorial of a non-negative integer using recursion
+ * with memoization (top-down dynamic programming). It stores intermediate results
+ * to avoid redundant calculations for improved efficiency.
+ * 
+ * Memoization is a form of caching where the result to an expensive function call
+ * is stored and returned.
+ * Example:
+ * Input: n = 5
+ * Output: 120
+ * 
+ * Explanation: 5! = 5 × 4 × 3 × 2 × 1 = 120
+ * 
+ * The program uses a recursive function fact_recursion which caches computed 
+ * results in a memo array to avoid recalculating factorials for the same numbers.
+ *
+ * Time Complexity: O(n)
+ * Space Complexity: O(n)
+ * @author [Vedant Mukhedkar](https://github.com/git5v)
+ */
+
+#include <iostream> // for std::cout
+#include <cassert>  // For test cases
+#include <array>    // For std::array
+#include <cstdint>  // For uint64_t
+
+/// Array to store computed factorials for memoization
+std::array<uint64_t, 1000> memo{0};
+
+/**
+ * @namespace math
+ * @brief Math algorithms
+ */
+namespace math {
+
+/**
+ * @brief Computes the factorial of a non-negative integer using recursion and memoization.
+ * @param n The integer whose factorial is to be computed
+ * @returns The factorial of n
+ */
+uint64_t fact_recursion(uint64_t n) {
+    if (n == 0) return 1;        // Base case: 0! = 1
+    if (memo[n] != 0) return memo[n]; // Return already computed value
+    memo[n] = n * fact_recursion(n - 1);    // Store and return the computed value
+    return memo[n];
+}
+
+} // namespace math
+/**
+ * @brief Self-test implementations for the fact_recursion function.
+ * @returns void
+ */
+void test_fact_recursion() {
+    // Test cases for factorial computation
+    assert(math::fact_recursion(0) == 1);
+    assert(math::fact_recursion(1) == 1);
+    assert(math::fact_recursion(5) == 120);
+    assert(math::fact_recursion(10) == 3628800);
+    std::cout << "All test cases passed!\n";
+}
+
+/**
+ * @brief Main function to run test cases and interact with the user.
+ * @returns 0 on program success
+ */
+int main() {
+    // Run test cases
+    test_fact_recursion();
+    return 0;
+}