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)


library(gmp)
 
left_factorial <- function(n) {
	if (n == 0) return(0)
	result <- as.bigz(0)
 
	adder <- as.bigz(1)	
	for (k in 1:n) {
		result <- result + adder
		adder <- adder * k
	}
	result
}
 
digit_count <- function(n) {
	nchar(as.character(n))
}
 
for (n in 0:10) {
	cat("!",n," = ",sep = "")
	cat(as.character(left_factorial(n)))
	cat("\n")
}
for (n in seq(20,110,10)) {
	cat("!",n," = ",sep = "")
	cat(as.character(left_factorial(n)))
	cat("\n")
}
for (n in seq(1000,10000,1000)) {
  cat("!",n," has ",digit_count(left_factorial(n))," digits\n", sep = "")
}
 