I was practising C. k-Tree. I want to solve it by finding the difference between f(n,k) and f(n,d-1) where f will return the number of ways to break n into a sum of integers from 1 to k.

Can someone please link the resources for an iterative solution or explain it?

Thank you!

Edit: Making it less confusing:

Example:

I want to find the number of ways to make sum n=3 from using integers 1 to k=2.

1st way: 1 + 1 + 1

2nd way: 2 + 1

3rd way: 1 + 2

So total ways are 3.

Note: I can not use only 3 because it is more than k=2. If k=3 then I can also use only 3 and get 4 ways.

I can do this by recursion like this:

ll recFun(ll k, ll n, ll sum = 0) {
  if (sum == n) {
    return 1;
  } else if (sum > n) return 0;
 
 
  ll total = 0;
 
  FO(i, k) {
    total += recFun(k, n, sum+i+1);
  }
 
  return total;
}