Left factorials

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) use the notations: !n` or $!n'$ or n¡.

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

!n=\sum _{k=0}^{n-1}k!

where

$!0=0$
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, 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]

D

<lang d>import std.stdio, std.bigint, std.range, std.algorithm, std.conv;

BigInt leftFact(in uint n) pure /*nothrow*/ {

   BigInt result = 0, factorial = 1;
   foreach (immutable i; 1 .. n + 1) {
       result += factorial;
       factorial *= i;
   }
   return result;

}

void main() {

   writeln("First 11 left factorials:\n", 11.iota.map!leftFact);
   writefln("\n20 through 110 (inclusive) by tens:\n%(%s\n%)",
            iota(20, 111, 10).map!leftFact);
   writefln("\nDigits in 1,000 through 10,000 by thousands:\n%s",
            iota(1000,10001, 1000).map!(i => i.leftFact.text.length));

}</lang>

Output:

First 11 left factorials:
[0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114]

20 through 110 (inclusive) by tens:
128425485935180314
9157958657951075573395300940314
20935051082417771847631371547939998232420940314
620960027832821612639424806694551108812720525606160920420940314
141074930726669571000530822087000522211656242116439949000980378746128920420940314
173639511802987526699717162409282876065556519849603157850853034644815111221599509216528920420940314
906089587987695346534516804650290637694024830011956365184327674619752094289696314882008531991840922336528920420940314
16695570072624210767034167688394623360733515163575864136345910335924039962404869510225723072235842668787507993136908442336528920420940314
942786239765826579160595268206839381354754349601050974345395410407078230249590414458830117442618180732911203520208889371641659121356556442336528920420940314
145722981061585297004706728001906071948635199234860720988658042536179281328615541936083296163475394237524337422204397431927131629058103519228197429698252556442336528920420940314

Digits in 1,000 through 10,000 by thousands:
[2565, 5733, 9128, 12670, 16322, 20062, 23875, 27749, 31678, 35656]

Perl 6

Perl 6 doesn't have a built in factorial function, so the first two lines implement postfix ! factorial. The newly implemented factorial function is used to implement left factorial using a prefix ! in the next two lines. Note that this redefines the core prefix ! (not) function. The last two lines are display code for the various sub task requirements.

<lang perl6>multi sub postfix:<!> (0) { 1 }; multi sub postfix:<!> ($n) { [*] 1 .. $n }; multi sub prefix:<!> (0) { 0 }; multi sub prefix:<!> ($k) { [+] (^$k).map: { $_! } }

printf "!%d = %s\n", $_, !$_ for ^11, 20, 30 ... 110; printf "!%d has %d digits.\n", $_, (!$_).chars for 1000, 2000 ... 10000;</lang>

Output:

!0  = 0
!1  = 1
!2  = 2
!3  = 4
!4  = 10
!5  = 34
!6  = 154
!7  = 874
!8  = 5914
!9  = 46234
!10  = 409114
!20  = 128425485935180314
!30  = 9157958657951075573395300940314
!40  = 20935051082417771847631371547939998232420940314
!50  = 620960027832821612639424806694551108812720525606160920420940314
!60  = 141074930726669571000530822087000522211656242116439949000980378746128920420940314
!70  = 173639511802987526699717162409282876065556519849603157850853034644815111221599509216528920420940314
!80  = 906089587987695346534516804650290637694024830011956365184327674619752094289696314882008531991840922336528920420940314
!90  = 16695570072624210767034167688394623360733515163575864136345910335924039962404869510225723072235842668787507993136908442336528920420940314
!100  = 942786239765826579160595268206839381354754349601050974345395410407078230249590414458830117442618180732911203520208889371641659121356556442336528920420940314
!110  = 145722981061585297004706728001906071948635199234860720988658042536179281328615541936083296163475394237524337422204397431927131629058103519228197429698252556442336528920420940314
!1000 has 2565 digits.
!2000 has 5733 digits.
!3000 has 9128 digits.
!4000 has 12670 digits.
!5000 has 16322 digits.
!6000 has 20062 digits.
!7000 has 23875 digits.
!8000 has 27749 digits.
!9000 has 31678 digits.
!10000 has 35656 digits.

Python

<lang python>from itertools import islice

def lfact():

   yield 0
   fact, summ, n = 1, 0, 1 
   while 1:
       fact, summ, n = fact*n, summ + fact, n + 1
       yield summ

print('first 11:\n %r' % [lf for i, lf in zip(range(11), lfact())]) print('20 through 110 (inclusive) by tens:') for lf in islice(lfact(), 20, 111, 10):

   print(lf)

print('Digits in 1,000 through 10,000 (inclusive) by thousands:\n %r'

     % [len(str(lf)) for lf in islice(lfact(), 1000, 10001, 1000)] )</lang>

Output:

first 11:
  [0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114]
20 through 110 (inclusive) by tens:
128425485935180314
9157958657951075573395300940314
20935051082417771847631371547939998232420940314
620960027832821612639424806694551108812720525606160920420940314
141074930726669571000530822087000522211656242116439949000980378746128920420940314
173639511802987526699717162409282876065556519849603157850853034644815111221599509216528920420940314
906089587987695346534516804650290637694024830011956365184327674619752094289696314882008531991840922336528920420940314
16695570072624210767034167688394623360733515163575864136345910335924039962404869510225723072235842668787507993136908442336528920420940314
942786239765826579160595268206839381354754349601050974345395410407078230249590414458830117442618180732911203520208889371641659121356556442336528920420940314
145722981061585297004706728001906071948635199234860720988658042536179281328615541936083296163475394237524337422204397431927131629058103519228197429698252556442336528920420940314
Digits in 1,000 through 10,000 (inclusive) by thousands:
  [2565, 5733, 9128, 12670, 16322, 20062, 23875, 27749, 31678, 35656]

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

Ruby

<lang ruby>left_fact = Enumerator.new do |y|

 n, f, lf = 0, 1, 0
 loop do
   y  << lf #yield left_factorial
   n  += 1
   lf += f
   f  *= n
 end

end

tens = 20.step(110, 10).to_a thousands = 1000.step(10_000, 1000)

10001.times do |n|

 lf = left_fact.next
 case n
 when 0..10, *tens
   puts "!#{n} = #{lf}"
 when *thousands
   puts "!#{n} has #{lf.to_s.size} digits"
 end

end </lang>

Output:

!0 = 0
!1 = 1
!2 = 2
!3 = 4
!4 = 10
!5 = 34
!6 = 154
!7 = 874
!8 = 5914
!9 = 46234
!10 = 409114
!20 = 128425485935180314
!30 = 9157958657951075573395300940314
!40 = 20935051082417771847631371547939998232420940314
!50 = 620960027832821612639424806694551108812720525606160920420940314
!60 = 141074930726669571000530822087000522211656242116439949000980378746128920420940314
!70 = 173639511802987526699717162409282876065556519849603157850853034644815111221599509216528920420940314
!80 = 906089587987695346534516804650290637694024830011956365184327674619752094289696314882008531991840922336528920420940314
!90 = 16695570072624210767034167688394623360733515163575864136345910335924039962404869510225723072235842668787507993136908442336528920420940314
!100 = 942786239765826579160595268206839381354754349601050974345395410407078230249590414458830117442618180732911203520208889371641659121356556442336528920420940314
!110 = 145722981061585297004706728001906071948635199234860720988658042536179281328615541936083296163475394237524337422204397431927131629058103519228197429698252556442336528920420940314
!1000 has 2565 digits
!2000 has 5733 digits
!3000 has 9128 digits
!4000 has 12670 digits
!5000 has 16322 digits
!6000 has 20062 digits
!7000 has 23875 digits
!8000 has 27749 digits
!9000 has 31678 digits
!10000 has 35656 digits

Tcl

<lang tcl>proc leftfact {n} {

   set s 0
   for {set i [set f 1]} {$i <= $n} {incr i} {

incr s $f set f [expr {$f * $i}]

   }
   return $s

}

for {set i 0} {$i <= 110} {incr i [expr {$i>9?10:1}]} {

   puts "!$i = [leftfact $i]"

} for {set i 1000} {$i <= 10000} {incr i 1000} {

   puts "!$i has [string length [leftfact $i]] digits"

}</lang>

Output:

!0 = 0
!1 = 1
!2 = 2
!3 = 4
!4 = 10
!5 = 34
!6 = 154
!7 = 874
!8 = 5914
!9 = 46234
!10 = 409114
!20 = 128425485935180314
!30 = 9157958657951075573395300940314
!40 = 20935051082417771847631371547939998232420940314
!50 = 620960027832821612639424806694551108812720525606160920420940314
!60 = 141074930726669571000530822087000522211656242116439949000980378746128920420940314
!70 = 173639511802987526699717162409282876065556519849603157850853034644815111221599509216528920420940314
!80 = 906089587987695346534516804650290637694024830011956365184327674619752094289696314882008531991840922336528920420940314
!90 = 16695570072624210767034167688394623360733515163575864136345910335924039962404869510225723072235842668787507993136908442336528920420940314
!100 = 942786239765826579160595268206839381354754349601050974345395410407078230249590414458830117442618180732911203520208889371641659121356556442336528920420940314
!110 = 145722981061585297004706728001906071948635199234860720988658042536179281328615541936083296163475394237524337422204397431927131629058103519228197429698252556442336528920420940314
!1000 has 2565 digits
!2000 has 5733 digits
!3000 has 9128 digits
!4000 has 12670 digits
!5000 has 16322 digits
!6000 has 20062 digits
!7000 has 23875 digits
!8000 has 27749 digits
!9000 has 31678 digits
!10000 has 35656 digits