Palindromes

A string is called a palindrome if it reads the same from left to right and from right to left. For example, the string "abba" is a palindrome, while "omax" is not. A substring of string $\alpha$ from position $i$ to position $j$ inclusive (positions are numbered from $1$) is denoted as $\alpha [i..j]$. The length of such a substring is j - i + 1.

For the given string $\alpha$ of length $n$, find the number $q$ of pairs $(i,j)$ such that 1 \le i < j \le n and the substring $\alpha [i..j]$ is a palindrome.