Left factorials,   !n,   may refer to either   subfactorials   or to   factorial sums;

the same notation can be confusingly seen being used for the two different definitions.

Sometimes,   subfactorials   (also known as derangements)   may use any of the notations:

     !n`   
     !n  
     n¡  


(It may not be visually obvious, but the last example uses an upside-down exclamation mark.)



This Rosetta Code task will be using this formula   (factorial sums)   for   left factorial:


   



!
n
=

∑

k
=
0


n
−
1


k
!


{\displaystyle !n=\sum _{k=0}^{n-1}k!}




 where


   



!
0
=
0


{\displaystyle !0=0}






Task
Display the left factorials for:

   zero through ten     (inclusive)
   20   through   110   (inclusive)   by tens


Display the length (in decimal digits) of the left factorials for:

   1,000   through   10,000   (inclusive), by thousands.


Also see

   The OEIS entry: A003422 left factorials
   The MathWorld entry: left factorial
   The MathWorld entry: factorial sums
   The MathWorld entry: subfactorial


Related task

   permutations/derangements (subfactorials)


import java.math.BigInteger;
 
public class LeftFac{
	public static BigInteger factorial(BigInteger n){
		BigInteger ans = BigInteger.ONE;
		for(BigInteger x = BigInteger.ONE; x.compareTo(n) <= 0; x = x.add(BigInteger.ONE)){
			ans = ans.multiply(x);
		}
		return ans;
	}
 
	public static BigInteger leftFact(BigInteger n){
		BigInteger ans = BigInteger.ZERO;
		for(BigInteger k = BigInteger.ZERO; k.compareTo(n.subtract(BigInteger.ONE)) <= 0; k = k.add(BigInteger.ONE)){
			ans = ans.add(factorial(k));
		}
		return ans;
	}
 
	public static void main(String[] args){
		for(int i = 0; i <= 10; i++){
			System.out.println("!" + i + " = " + leftFact(BigInteger.valueOf(i)));
		}
 
		for(int i = 20; i <= 110; i += 10){
			System.out.println("!" + i + " = " + leftFact(BigInteger.valueOf(i)));
		}
 
		for(int i = 1000; i <= 10000; i += 1000){
			System.out.println("!" + i + " has " + leftFact(BigInteger.valueOf(i)).toString().length() + " digits");
		}
	}
}