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)


      INSTALL @lib$+"BB4WMAPMLIB" : PROCMAPM_Init : MAPM_Dec%=200
 
      Result$="0" : A$="1"
      FOR I%=0 TO 10000
        IF I% Result$=FNMAPM_Add(Result$,A$) : A$=FNMAPM_Multiply(A$,STR$I%)
 
        IF I% < 111 IF I% MOD 10 = 0 OR I% < 11 PRINT "!";I% " = " FNMAPM_FormatDec(Result$,0)
 
        IF I% > 999 IF I% MOD 1000 = 0 PRINT "!";I% " has " LENFNMAPM_FormatDec(Result$,0) " digits"
      NEXT
      END