<p>You are given <code>n</code> balloons, indexed from <code>0</code> to <code>n - 1</code>. Each balloon is painted with a number on it represented by an array <code>nums</code>. You are asked to burst all the balloons.</p>

<p>If you burst the <code>i<sup>th</sup></code> balloon, you will get <code>nums[i - 1] * nums[i] * nums[i + 1]</code> coins. If <code>i - 1</code> or <code>i + 1</code> goes out of bounds of the array, then treat it as if there is a balloon with a <code>1</code> painted on it.</p>

<p>Return <em>the maximum coins you can collect by bursting the balloons wisely</em>.</p>

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

<pre>
<strong>Input:</strong> nums = [3,1,5,8]
<strong>Output:</strong> 167
<strong>Explanation:</strong>
nums = [3,1,5,8] --&gt; [3,5,8] --&gt; [3,8] --&gt; [8] --&gt; []
coins =  3*1*5    +   3*5*8   +  1*3*8  + 1*8*1 = 167</pre>

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

<pre>
<strong>Input:</strong> nums = [1,5]
<strong>Output:</strong> 10
</pre>

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

<ul>
	<li><code>n == nums.length</code></li>
	<li><code>1 &lt;= n &lt;= 300</code></li>
	<li><code>0 &lt;= nums[i] &lt;= 100</code></li>
</ul>

import numpy as np

class Solution:
    
    def recurse(self, nums, start, end):
        if self.dp[start, end] >= 0:
            return self.dp[start, end]
        
        N = len(nums)
        
        outer_prod = (nums[start-1] if start != 0 else 1) * (nums[end] if end != N  else 1)
        
        if start == end:
            ret = 0
        else:
            max_value = 0
            for i in range(start, end):
                value = self.recurse(nums, start, i) + self.recurse(nums, i+1, end) + outer_prod * nums[i]
                if value > max_value:
                    max_value = value
            ret = max_value
        
        self.dp[start, end] = ret
        
        #print(start, end, ret)
        return ret
        
    def maxCoins(self, nums: List[int]) -> int:
        N = len(nums)
        self.dp = np.zeros((N+1, N+1), dtype=int)
        self.dp[:] = -1
        return self.recurse(nums, 0, N)

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