1. If there is only one character then f(a,a) = 1
2. If there is a substring starting from index a to index b then,
   1. If str[a] and str[b] both are same then
      f(a,b)=f(a+1,b)+f(a,b-1)+1
   2. If str[a] and str[b] both are not same then
      f(a,b)=f(a+1,b)+f(a,b-1)-f(a+1,b-1)

For, str = abbaa

From the above, it is understandable that one function is called repeatedly so for the large input the repetition will be very high. Because of this problem we are using a Dynamic programming approach to avoid repetition. We are using the memorization process to solve the problem.

Problem solution:

Recursive algorithm:

int Function(str,pos,l):
	if str.length()== l
		return
	for a=pos to str.length()-l 
		if str[a]==str[a+l-1]
			return Function(str,a,l-1)+Function(str,a+1,l-1)+1
		else
			return Function(str,a,l-1)+Function(str,a+1,l-1)-Function(str,a+1,l-2)

DP conversion:

int Function(str,n):
	for len=1 to str.length()
		for a=0 to str.length()-len
			b=pos+len-1
			if str[a]==str[b]
				arr[a][b]=arr[a+1][b]+ arr[a][b-1]+1
			else
				arr[a][b]=arr[a+1][b]+arr[a][b-1]-arr[a+1][b-1]
	return arr[0][len-1]

C++ Implementation:

#include <bits/stdc++.h>
using namespace std;

int count_palindrome(string str)
{
    int len = str.length();
    int arr[len][len] = { 0 };
    for (int i = 0; i < len; i++) {
        arr[i][i] = 1;
    }
    for (int l = 2; l <= len; l++) {
        for (int i = 0; i <= len - l; i++) {
            int j = i + l - 1;
            if (str[i] == str[j]) {
                arr[i][j] = arr[i + 1][j] + arr[i][j - 1] + 1;
            }
            else {
                arr[i][j] = arr[i + 1][j] + arr[i][j - 1] - arr[i + 1][j - 1];
            }
        }
    }
    return arr[0][len - 1];
}

int main()
{
    int t;
    cout << "Test Case : ";
    cin >> t;
    while (t--) {
        cout << "Enter the string : ";
        string str;
        cin >> str;
        cout << "Number of palindromic subsequences are : " << count_palindrome(str) << endl;
    }

    return 0;
}

Output

Test Case : 3
Enter the string : abc
Number of palindromic subsequences are : 3
Enter the string : aaa
Number of palindromic subsequences are : 7
Enter the string : abcc
Number of palindromic subsequences are : 5