<p>Given a string s, return <em>the number of different non-empty palindromic subsequences in</em> <code>s</code>. Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>A subsequence of a string is obtained by deleting zero or more characters from the string.</p>

<p>A sequence is palindromic if it is equal to the sequence reversed.</p>

<p>Two sequences <code>a<sub>1</sub>, a<sub>2</sub>, ...</code> and <code>b<sub>1</sub>, b<sub>2</sub>, ...</code> are different if there is some <code>i</code> for which <code>a<sub>i</sub> != b<sub>i</sub></code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;bccb&quot;
<strong>Output:</strong> 6
<strong>Explanation:</strong> The 6 different non-empty palindromic subsequences are &#39;b&#39;, &#39;c&#39;, &#39;bb&#39;, &#39;cc&#39;, &#39;bcb&#39;, &#39;bccb&#39;.
Note that &#39;bcb&#39; is counted only once, even though it occurs twice.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;abcdabcdabcdabcdabcdabcdabcdabcddcbadcbadcbadcbadcbadcbadcbadcba&quot;
<strong>Output:</strong> 104860361
<strong>Explanation:</strong> There are 3104860382 different non-empty palindromic subsequences, which is 104860361 modulo 10<sup>9</sup> + 7.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length &lt;= 1000</code></li>
	<li><code>s[i]</code> is either <code>&#39;a&#39;</code>, <code>&#39;b&#39;</code>, <code>&#39;c&#39;</code>, or <code>&#39;d&#39;</code>.</li>
</ul>

class Solution:

    def __init__(self):
        self.cache = {}
    
    def f(self, s, i, j):
        if i > j:
            return 0
        if i == j:
            return 1
        if (i,j) in self.cache:
            return self.cache[(i,j)]
        
        if s[i] == s[j]:
            low, high = i+1, j-1
            while low <= high and s[low] != s[i]:
                low += 1
            while low <= high and s[high] != s[j]:
                high -= 1
            
            if low > high:
                res = self.f(s,i+1, j-1) * 2 + 2
            elif low == high:
                res = self.f(s, i+1, j-1) * 2 + 1
            else:
                res = self.f(s, i+1, j-1) * 2 - self.f(s,low+1,high-1)
        else:
            res = self.f(s, i, j-1) + self.f(s, i+1, j) - self.f(s, i+1, j-1)
        
        self.cache[(i,j)] = res % 1000000007
        return self.cache[(i,j)]

    def countPalindromicSubsequences(self, s: str) -> int:
        return self.f(s,0,len(s)-1)


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