Longest Palindromic Substring

Find the longest substring that reads the same forwards and backwards. Expand around centres: every palindrome has a centre that's either a single character (odd length) or between two characters (even length). There are $2n - 1$ centres, and expanding each costs up to $O(n)$, giving $O(n^2)$ overall in $O(1)$ extra space.

Implement longest_palindrome(s), returning the first-occurring longest one on ties ("" for empty input). Then implement count_palindromic_substrings(s): how many substrings (counted by position) are palindromes.