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)


lf: procedure (n) returns (fixed decimal (31) );
   declare n fixed binary;
   declare (s, f) fixed (31);
   declare (i, j) fixed;
 
   s = 0;
   do i = n-1 to 0 by -1;
      f = 1;
      do j = i to 1 by -1;
         f = f * j;
      end;
      s = s + f;
   end;
   return (s);
end lf;
 
   declare n fixed binary;
 
   do n = 0 to 10, 20 to 30;
      put skip list ('Left factorial of ' || n || '=' || lf(n) );
   end;
 
end left_factorials;