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)


proc leftfact {n} {
    set s 0
    for {set i [set f 1]} {$i <= $n} {incr i} {
	incr s $f
	set f [expr {$f * $i}]
    }
    return $s
}
 
for {set i 0} {$i <= 110} {incr i [expr {$i>9?10:1}]} {
    puts "!$i = [leftfact $i]"
}
for {set i 1000} {$i <= 10000} {incr i 1000} {
    puts "!$i has [string length [leftfact $i]] digits"
}