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)


 
#!/usr/bin/gawk -Mf
function left_factorial(num) {
	result=0
	adder=1
	if (num==0) return(0)
	for (k = 1; k <=num; k++) {
		result = result + adder
		adder = adder * k
	}
	return(result)
}
 
BEGIN {
	for (i = 0; i <= 10; i++) {
		print "!" i " = " left_factorial(i)
	}
	for (i = 20; i<= 110; i+=10) {
		print "!" i " = " left_factorial(i)
	}
	for (i = 1000; i<= 10000; i+=1000) {
		print "!" i " has " length(left_factorial(i)) " digits"
	}
      }
 