DP — Count Palindromic Substrings

solve(s) — return how many substrings of s are palindromes. Substrings with
different positions count separately, even when their text is identical.

Same idea as the longest palindromic substring, but every successful expansion step
adds one more palindrome to the count.

Examples

solve("abc")      → 3     // "a", "b", "c"
solve("aaa")      → 6     // "a" ×3, "aa" ×2, "aaa"
solve("abba")     → 6     // "a", "b", "b", "a", "bb", "abba"
solve("racecar")  → 10