111. Count palindromic substrings

A palindromic string reads the same backward as forward, such as 'madam'.

Now given a string, count how many substrings it has?

Like 'madam', it has following palindromic strings:

'm''a''d''a''m''ada''madam'

What is the time and space cost of your solution ? Could you improve it ?

Thanks to @TechieQian for helping with the test cases.