Sum of factorials of 1 to n using only one recursive function

Question

I need a single recursive function to compute the series S=1! + 2! + 3! + ...+n!.

I've done it using 2 functions.

void main()
{ 
  int terms=4;
  printf(sum(terms));  //outputs 33
 /* 
    terms is the number of terms in the series
    e.g. terms=3 outputs 9, 4 outputs 33
 */
}

int fac(int n)   
{            
     return n<2 ? 1 : n * fac(n-1) ;
}   
int sum(int terms)   
{     
     return terms>0 ? (fac(terms) + sum(terms-1)):0 ;
}

Answer 1

This is the simplest I could get it:

int sum_fac(int target, int counter) {
    return (target == counter) ? target : counter * (1 + sum_fac(target, counter + 1));
}

Which when called like this:

int main() {
  for (int i = 1; i < 10; i++) {
    printf("%d: %d\n", i, sum_fac(i, 1));
  }
}

Outputs this:

1: 1
2: 3
3: 9
4: 33
5: 153
6: 873
7: 5913
8: 46233
9: 409113