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 "/fmt" for Fmt
import "/big" for BigInt
 
var lfacts = List.filled(12, BigInt.zero)
var lfact = BigInt.one
var sum = BigInt.zero
 
for (i in 1..10) {
    sum = sum + lfact
    lfacts[i] = sum
    lfact = lfact * i
}
System.print("Left factorials from 0 to 10:")
for (i in 0..10) System.write(" %(lfacts[i])")
 
for (i in 11..110) {
    sum = sum + lfact
    if (i%10 == 0) lfacts[i/10] = sum
    lfact = lfact * i
}
System.print("\n\nLeft factorials from 20 to 110 by tens:")
for (i in 2..11) Fmt.print(" !$-3d -> $s", i * 10, lfacts[i])
 
for (i in 111..10000) {
    sum = sum + lfact
    if (i%1000 == 0) lfacts[i/1000] = sum
    lfact = lfact * i
}
System.print("\nLengths of left factorals from 1000 to 10000 by thousands:")
for (i in 1..10) Fmt.print(" !$-5d -> $5s", i * 1000, lfacts[i].toString.count)