<p>You are given an integer array <code>nums</code> and an integer <code>k</code>. You can partition the array into <strong>at most</strong> <code>k</code> non-empty adjacent subarrays. The <strong>score</strong> of a partition is the sum of the averages of each subarray.</p>

<p>Note that the partition must use every integer in <code>nums</code>, and that the score is not necessarily an integer.</p>

<p>Return <em>the maximum <strong>score</strong> you can achieve of all the possible partitions</em>. Answers within <code>10<sup>-6</sup></code> of the actual answer will be accepted.</p>

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

<pre>
<strong>Input:</strong> nums = [9,1,2,3,9], k = 3
<strong>Output:</strong> 20.00000
<strong>Explanation:</strong> 
The best choice is to partition nums into [9], [1, 2, 3], [9]. The answer is 9 + (1 + 2 + 3) / 3 + 9 = 20.
We could have also partitioned nums into [9, 1], [2], [3, 9], for example.
That partition would lead to a score of 5 + 2 + 6 = 13, which is worse.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4,5,6,7], k = 4
<strong>Output:</strong> 20.50000
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 100</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
	<li><code>1 &lt;= k &lt;= nums.length</code></li>
</ul>

class Solution {
 public:
  double largestSumOfAverages(vector<int>& nums, int k) {
    const int n = nums.size();
    vector<vector<double>> mem(n + 1, vector<double>(k + 1));
    vector<double> prefix(n + 1);
    partial_sum(nums.begin(), nums.end(), prefix.begin() + 1);
    return largestSumOfAverages(nums, n, k, prefix, mem);
  }

 private:
  // Returns the maximum score to partition the first i numbers into k groups.
  double largestSumOfAverages(const vector<int>& nums, int i, int k,
                              const vector<double>& prefix,
                              vector<vector<double>>& mem) {
    if (k == 1)
      return prefix[i] / i;
    if (mem[i][k] > 0)
      return mem[i][k];

    // Try all the possible partitions.
    for (int j = k - 1; j < i; ++j)
      mem[i][k] =
          max(mem[i][k], largestSumOfAverages(nums, j, k - 1, prefix, mem) +
                             (prefix[i] - prefix[j]) / (i - j));

    return mem[i][k];
  }
};
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