Simple DP Solution using sort (rust, Javascript)
ApproachFirst, you need to count the subsequences based on the number of elements using dynamic programming (DP). If dp[l] represents the count of subsequences
# Approach
<!-- Describe your approach to solving the problem. -->
First, you need to count the subsequences based on the number of elements using dynamic programming (DP). If dp[l] represents the count of subsequences with length l, then adding a new element updates the count for length l + 1 as dp[l+1] += dp[l]. Additionally, you should add 1 to dp[0].

For caluculating maximum case, sort the array nums and calculate the subsequences by incrementally adding elements from nums in ascending order. When adding nums[i], the maximum value of the newly formed subsequences becomes nums[i]. Therefore, you need to add nums[i] multiplied by the count of already generated subsequences. The total count of subsequences is the sum of all values in dp.

Finally, compute the return value based on the minimum case.
# Complexity
- Time complexity:
<!-- Add your time complexity here, e.g. $$O(n)$$ -->
$$O(nums.len()*k)$$
- Space complexity:
<!-- Add your space complexity here, e.g. $$O(n)$$ -->
$$O(k)$$
# Code
```rust []
const MOD:i64 = 1000000007;
impl Solution {
    pub fn min_max_sums(nums: Vec<i32>, k: i32) -> i32 {
        let mut nums = nums;
        nums.sort();
        let mut ret: i64 = 0;
        let mut dp: Vec<i64> = vec![0;k as usize];
        for i in 0..nums.len() {
            let mut c:i64 = 1;
            for j in (1..dp.len()).rev() {
                c = (c+dp[j-1])%MOD;;
                dp[j] = (dp[j]+dp[j-1])%MOD;
            }
            dp[0] += 1;
            ret = (ret+c*nums[i] as i64)%MOD;
            ret = (ret+c*nums[nums.len()-1-i] as i64)%MOD;
        }
        return ret as i32;
    }
}
```
```Javascript []
var minMaxSums = function(nums, k) {
    const MOD = 1000000007
    const BMOD = BigInt(1000000007)
    nums.sort((a, b) => a - b) 
    let ret = 0
    const dp = Array(k).fill(0)
    for (let i = 0; i < nums.length; i += 1){
        let c = 1
        for(let j = dp.length-1; j > 0; j -= 1){
            c = (c+dp[j-1])%MOD;
            dp[j] = (dp[j]+dp[j-1])%MOD;
        }
        dp[0] += 1;
        ret = (ret+Number(BigInt(c)*BigInt(nums[i])%BMOD))%MOD;
        ret = (ret+Number(BigInt(c)*BigInt(nums[nums.length-1-i])%BMOD))%MOD
    }
    return ret
};
```

['Dynamic Programming', 'Rust', 'JavaScript']