Recursive solution of ordered Coin Combinations II (CSES) - c++

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Consider a money system consisting of n coins. Each coin has a positive integer value. Your task is to calculate the number of distinct ordered ways you can produce a money sum x using the available coins.
For example, if the coins are {2,3,5} and the desired sum is 9, there are 3 ways:
2+2+5
3+3+3
2+2+2+3
I am trying to write recursive solution for this.
I came up with this code.
#include<iostream>
#include<vector>
using namespace std;
#define M 1000000007
vector<int>arr;
vector<vector<int>>dp;
int solve(int i,int target){
if(i>=(int)arr.size()) return 0;
if(target<0) return 0;
if(target==0) return 1;
if(dp[i][target]!=-1) return dp[i][target];
return dp[i][target]=(solve(i,target-arr[i])%M+solve(i+1,target)%M)%M;
}
int main(){
int n,target;
cin>>n>>target;
arr.resize(n);
dp.resize(n+1,vector<int>(target+1,-1));
for(int i=0;i<n;++i){
cin>>arr[i];
}
cout<<solve(0,target);
}
but this code is giving time-limit-exceeded. How can this code be written in a recursive manner so that it gets accepted? Here is non-recursive approach but I want to know how to write a recursive version of this.

Answer 1

an example of recursive sum which select results would be:
void recursive_sum( IN const std::vector<int>& coins,
IN int result,
OUT std::vector<std::string>& res,
IN std::deque<int> numbers = {},
IN int actual_sum = 0)
{
for(auto& n: coins)
{
int nsum = actual_sum + n;
if(nsum == result)
insert_result(res, copy_insert(numbers, n));
else if(nsum < result)
recursive_sum(coins, result, res, copy_insert(numbers, n), nsum);
}
}
a main similar to this would do the job:
int main()
{
std::vector<std::string> res{};
std::vector<int> coins{};
int sum_result = 0;
input_coins(coins);
input_sum_result(sum_result);
recursive_sum(coins, sum_result, res);
out_result(res);
return 0;
}
if you want to take a look to a full version of this example click here.
output example:
"
Insert a number or 'next' to continue
10
Actually coins are: { 10, }
Insert a number or 'next' to continue
15
Actually coins are: { 10, 15, }
Insert a number or 'next' to continue
25
Actually coins are: { 10, 15, 25, }
Insert a number or 'next' to continue
next
Insert the sum result:
100
Possible sums are:
{10, 10, 10, 10, 10, 10, 10, 10, 10, 10, }
{10, 10, 10, 10, 10, 10, 10, 15, 15, }
{10, 10, 10, 10, 10, 10, 15, 15, 10, }
{10, 10, 10, 10, 10, 10, 15, 25, }
{10, 10, 10, 10, 10, 10, 25, 15, }
{10, 10, 10, 10, 10, 25, 10, 15, }
{10, 10, 10, 10, 10, 25, 25, }
{10, 10, 10, 10, 15, 15, 15, 15, }
{10, 10, 10, 15, 15, 10, 15, 15, }
{10, 10, 10, 15, 15, 15, 25, }
{10, 10, 10, 15, 25, 15, 15, }
{10, 10, 15, 15, 10, 25, 15, }
{10, 10, 15, 15, 15, 10, 10, 15, }
{10, 10, 15, 15, 15, 10, 25, }
{10, 10, 15, 15, 25, 25, }
{10, 15, 15, 15, 15, 15, 15, }
{10, 15, 15, 25, 10, 25, }
{10, 15, 25, 25, 10, 15, }
{10, 15, 25, 25, 25, }
{10, 25, 10, 10, 10, 10, 25, }
{10, 25, 10, 10, 15, 10, 10, 10, }
{15, 15, 15, 15, 15, 25, }
{15, 25, 15, 15, 15, 15, }
{25, 25, 25, 10, 15, }
{25, 25, 25, 15, 10, }
{25, 25, 25, 25, }
"