**Problem Statement:**
Given the required input arguments, write an efficient algorithm to solve the **Count Palindromic Substrings** problem. Implement the required logic as specified by standard definitions for this classic algorithmic challenge.
**Hint / Expected Approach:**
Expand around center, count each palindrome
**Edge Cases to Consider:**
- (1) Single character
- (2) All same characters (n(n+1)/2)
- (3) No palindromes > length 1