O(n) Palindromic Substring: Why is O(n^2) needed here?
Consider the following string
$$ dabacabac $$
When you go to compute $T(9)$ (using one indexing), you have $T(8) = 7$. But since 
$$ T(1) = d \neq c = T(9) $$
You will put $T(9) = 1$, when it should be $T(9) = 5$.