Left factorials

Left factorials is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Left factorials have a confusing name as it can refer to:

• subfactorials:     !n
• factorial sums:   !n

and one can see the exact notation being used for both.

Sometimes, subfactorials use the notation:   !n`   or   !n'   or   n¡.

(Subfactorials are also known as derangements.)

This Rosetta Code task will be using the formula for left factorial:

  $!n=\sum _{k=0}^{n-1}k!$

where

!0 = 0

task requirements

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] in The On-Line Encyclopedia of Integer Sequences (R).
• The MathWorld (TM) entry: [left factorial] in Wolfram MathWorld (TM).
• The MathWorld (TM) entry: [factorial sums] in Wolfram MathWorld (TM).
• The MathWorld (TM) entry: [subfactorial] in Wolfram MathWorld (TM).

REXX

<lang rexx>/*REXX pgm computes/shows the left factorial (or width) of N (or range).*/ parse arg bot top inc . /*obtain optional args from C.L. */ if bot== then bot=1 /*BOT defined? Then use default.*/ td= bot<0 /*if BOT < 0, only show # digs.*/ bot=abs(bot) /*use the |bot| for the DO loop.*/ if top== then top=bot /* " " top " " " " */ if inc= then inc=1 /* " " inc " " " " */ @='left ! of ' /*a literal used in the display. */ w=length(H) /*width of largest number request*/

          do j=bot  to top  by inc    /*traipse through  #'s requested.*/
          if td  then say @ right(j,w)  " ───► "  length(L!(j)) ' digits'
                 else say @ right(j,w)  " ───► "  L!(j)
          end   /*j*/                  /* [↑]  show either L! or #digits*/

exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────L! subroutine───────────────────────*/ L!: procedure; parse arg x .; if x<3 then return x; s=4 /*shortcuts.*/ !=2; do f=3 to x-1 /*compute L! for all numbers───►X*/

        !=!*f                         /*compute intermediate factorial.*/
        if pos(.,!)\==0 then numeric digits digits()*1.5%1 /*bump digs.*/
        s=s+!                         /*add the factorial ───► L!  sum.*/
        end   /*f*/                   /* [↑]  handles gi-hugeic numbers*/

return s /*return the sum (L!) to invoker.*/</lang> output when using the input:   0 10

left ! of   0  ───►  0
left ! of   1  ───►  1
left ! of   2  ───►  2
left ! of   3  ───►  4
left ! of   4  ───►  10
left ! of   5  ───►  34
left ! of   6  ───►  154
left ! of   7  ───►  874
left ! of   8  ───►  5914
left ! of   9  ───►  46234
left ! of  10  ───►  409114

output when using the input:   20 110 10

left ! of   20  ───►  128425485935180314
left ! of   30  ───►  9157958657951075573395300940314
left ! of   40  ───►  20935051082417771847631371547939998232420940314
left ! of   50  ───►  620960027832821612639424806694551108812720525606160920420940314
left ! of   60  ───►  141074930726669571000530822087000522211656242116439949000980378746128920420940314
left ! of   70  ───►  173639511802987526699717162409282876065556519849603157850853034644815111221599509216528920420940314
left ! of   80  ───►  906089587987695346534516804650290637694024830011956365184327674619752094289696314882008531991840922336528920420940314
left ! of   90  ───►  16695570072624210767034167688394623360733515163575864136345910335924039962404869510225723072235842668787507993136908442336528920420940314
left ! of  100  ───►  942786239765826579160595268206839381354754349601050974345395410407078230249590414458830117442618180732911203520208889371641659121356556442336528920420940314
left ! of  110  ───►  145722981061585297004706728001906071948635199234860720988658042536179281328615541936083296163475394237524337422204397431927131629058103519228197429698252556442336528920420940314

output when using the input:   -1000 10000 1000

left ! of   1000  ───►  2565  digits
left ! of   2000  ───►  5733  digits
left ! of   3000  ───►  9128  digits
left ! of   4000  ───►  12670  digits
left ! of   5000  ───►  16322  digits
left ! of   6000  ───►  20062  digits
left ! of   7000  ───►  23875  digits
left ! of   8000  ───►  27749  digits
left ! of   9000  ───►  31678  digits
left ! of  10000  ───►  35656  digits