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 iterutils, bigints
 
proc lfact: iterator: BigInt =
  result = iterator: BigInt =
    yield 0.initBigInt
    var
      fact = 1.initBigInt
      sum = 0.initBigInt
      n = 1.initBigInt
    while true:
      sum += fact
      fact *= n
      n += 1
      yield sum
 
echo "first 11:\n  "
for i in lfact().slice(last = 10):
  echo "  ", i
 
echo "20 through 110 (inclusive) by tens:"
for i in lfact().slice(20, 110, 10):
  echo "  ", i
 
echo "Digits in 1,000 through 10,000 (inclusive) by thousands:"
for i in lfact().slice(1_000, 10_000, 1_000):
  echo "  ", ($i).len