Left factorials

This page is a solution to the task Left factorials in the Rosetta Code, written in Fōrmulæ.

Description (from Rosetta Code)

Left factorials, !n, may refer to either subfactorials or to factorial sums; The same notation can be confusingly seen 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 for left factorial: where 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, 2,000 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)

Solution

The definition can perfectly work:

However, the following is much faster:

Case 1

Showing left factorials from zero to ten:

Case 2

Showing left factorials from 20 to 110, by tens:

Case 3

Showing length of left factorials, from 1,000 to 10,000 by thousands: