Total number of odd length palindrome sub-sequence around each center

Given a string str, the task is to find the number of odd length palindromic sub-sequences around of str with str[i] as center i.e. every index will be considered as the center one by one.

Examples: 

Input: str = "xyzx" 
Output: 1 2 2 1 
For index 0: There is only a single sub-sequence possible i.e. "x" 
For index 1: Two sub-sequences are possible i.e. "y" and "xyx" 
For index 2: "z" and "xzx" 
For index 3: "x"

Input: str = "aaaa" 
Output: 1 3 3 1 

Approach: We will use dynamic programming to solve this problem. Let's denote length of the string str be N. Now, Let dp[i][j] denote the number of palindromic sub-sequences from 0 to i - 1 and number of palindromic sub-sequences from j to N - 1. 
Let len be the distance between i and j. For each length len, we will fix our i and j, and check whether characters str[i] and str[j] are equal or not. Then according to it, we will make our dp transitions. 

If str[i] != str[j] then dp[i][j] = dp[i - 1][j] + dp[i][j + 1] - dp[i - 1][j + 1] 
If str[i] == str[j] then dp[i][j] = dp[i - 1][j] + dp[i][j + 1]
Base case: 
If i == 0 and j == n - 1 then dp[i][j] = 2 if str[i] == str[j] else dp[i][j] = 1. 
 

Below is the implementation of the above approach:  

1 3 4 3 1

Time Complexity: O(n²)
Auxiliary Space: O(n²), where n is the length of the given string.