Palindrome Factory

Consider an arbitrary string $\text{g}$. We will refer to this string as a palindrome generator. The set of palindromes $\text{P(g)}$ generated by this string is defined as follows:

Let the length of the string be $n$. For each $i$ from $1$ to $n$, the strings g[1..i]g[1..i]^r and g[1..i]g[1..i-1]^r are included in $\text{P(g)}$, where \alpha^r denotes the reverse of $\alpha$.

For example, if g = \text{olymp}, then P(g) = \{\text{oo}, \text{o}, \text{ollo}, \text{olo}, \text{olyylo}, \text{olylo}, \text{olymmylo}, \text{olymylo}, \text{olymppmylo}, \text{olympmylo}\}.

Given a palindrome generator $g$ and a string $s$, determine how many times strings from $P(g)$ appear as substrings in $s$. Specifically, find the number of pairs $(i,j)$ such that s[i..j] \in P(g).