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)


leftfact(N):-
    leftfact(N, 0, 0, 1).
 
leftfact(N, N, _, _):-
    !.
leftfact(N, M, L, F):-
    ((M =< 10 ; (M =< 110, 0 is M mod 10)) ->
        writef("!%w = %w\n", [M, L])
        ;
        (0 is M mod 1000 ->
            number_string(L, S),
            string_length(S, Len),
            writef("length of !%w is %w\n", [M, Len])
            ;
            true)),
    L1 is L + F,
    M1 is M + 1,
    F1 is F * M1,
    leftfact(N, M1, L1, F1).
 
main:-
    leftfact(10001).