3082. Find the Sum of the Power of All Subsequences

Problem Link

Description

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You are given an integer array nums of length n and a positive integer k.

The power of an array of integers is defined as the number of subsequences with their sum equal to k.

Return the sum of power of all subsequences of nums.

Since the answer may be very large, return it modulo 109 + 7.

 

Example 1:

Input: nums = [1,2,3], k = 3

Output: 6

Explanation:

There are 5 subsequences of nums with non-zero power:

• The subsequence [1,2,3] has 2 subsequences with sum == 3: [1,2,3] and [1,2,3].
• The subsequence [1,2,3] has 1 subsequence with sum == 3: [1,2,3].
• The subsequence [1,2,3] has 1 subsequence with sum == 3: [1,2,3].
• The subsequence [1,2,3] has 1 subsequence with sum == 3: [1,2,3].
• The subsequence [1,2,3] has 1 subsequence with sum == 3: [1,2,3].

Hence the answer is 2 + 1 + 1 + 1 + 1 = 6.

Example 2:

Input: nums = [2,3,3], k = 5

Output: 4

Explanation:

There are 3 subsequences of nums with non-zero power:

• The subsequence [2,3,3] has 2 subsequences with sum == 5: [2,3,3] and [2,3,3].
• The subsequence [2,3,3] has 1 subsequence with sum == 5: [2,3,3].
• The subsequence [2,3,3] has 1 subsequence with sum == 5: [2,3,3].

Hence the answer is 2 + 1 + 1 = 4.

Example 3:

Input: nums = [1,2,3], k = 7

Output: 0

Explanation: There exists no subsequence with sum 7. Hence all subsequences of nums have power = 0.

 

Constraints:

• 1 <= n <= 100
• 1 <= nums[i] <= 104
• 1 <= k <= 100

Solution

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Python3

class Solution:
    def sumOfPower(self, nums: List[int], k: int) -> int:
        N = len(nums)
        MOD = 10 ** 9 + 7
 
        @cache
        def go(index, ssum, count):
            if ssum == k: return pow(2, N - count, MOD)
 
            if ssum > k or index >= N: return 0
 
            skip = go(index + 1, ssum, count)
            take = go(index + 1, ssum + nums[index], count + 1)
 
            return (skip + take) % MOD
        
        return go(0, 0, 0)