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)


package main
 
import (
    "fmt"
    "math/big"
)
 
func main() {
    fmt.Print("!0 through !10: 0")
    one := big.NewInt(1)
    n := big.NewInt(1)
    f := big.NewInt(1)
    l := big.NewInt(1)
    next := func() { f.Mul(f, n); l.Add(l, f); n.Add(n, one) }
    for ; ; next() {
        fmt.Print(" ", l)
        if n.Int64() == 10 {
            break
        }
    }
    fmt.Println()
    for {
        for i := 0; i < 10; i++ {
            next()
        }
        fmt.Printf("!%d: %d\n", n, l)
        if n.Int64() == 110 {
            break
        }
    }
    fmt.Println("Lengths of !1000 through !10000 by thousands:")
    for i := 110; i < 1000; i++ {
        next()
    }
    for {
        fmt.Print(" ", len(l.String()))
        if n.Int64() == 10000 {
            break
        }
        for i := 0; i < 1000; i++ {
            next()
        }
    }
    fmt.Println()
}