bcDouble_Fact() function - PHPScience Labs

PHPSL Function: bcDouble_Fact( )

bcDouble_Fact ($IntN)

Dependencies: NONE

This function computes the double factorial value for a positive integer argument using arbitrary-precision. The double factorial of N is defined as the product of every positive integer greater than of equal to 1 with the same parity as N in the range from 1 up to and including N. The parity simply refers to whether an integer is an odd number or an even number.

If (N) is even, then the double factorial is the product of 2 and all the other even integers up to (N). If (N) is odd, then the double factorial is the product of 1 and all the other odd integers up to (N). See the examples below.

The double factorial values of 0 and −1 are equated to 1 by definition, but double factorial values for integers that are less than −1 are not defined.

The function returns FALSE on ERROR if (N) is non-numeric or (N < −1).

\begin{aligned} N!! = \prod_{n = 0}^{⌈ \frac{N}{2} ⌉ - 1} (N - 2 n) = N \cdot (N - 2) \cdot (N - 4) \cdot (N - 6) \cdot . . . \end{aligned}

Below is the PHP source code for a function to compute the arbitrary-precision double factorial.

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   function bcDouble_Fact ($IntN)
{
   $N = trim($IntN);   if (!is_numeric($N) or $N < -1) {return FALSE;}
   $j = ceil($N/2) - 1;
   $P = 1;  for($n=0;  $n <= $j;  $n++) {$P = bcmul($P, $N - 2*$n);}

   return $P;
}

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The above equation is really two equations in one.

For even numbers:

\begin{aligned} N!! = \prod_{n = 1}^{\frac{N}{2}} (2 n) = N \cdot (N - 2) \cdot (N - 4) \cdot (N - 6) \cdot . . . \cdot (6) \cdot (4) \cdot (2) \end{aligned}

12!! = 2 \cdot 4 \cdot 6 \cdot 8 \cdot 10 \cdot 12 = 46080

For odd numbers:

\begin{aligned} N!! = \prod_{n = 1}^{\frac{N + 1}{2}} (2 n - 1) = N \cdot (N - 2) \cdot (N - 4) \cdot (N - 6) \cdot . . . \cdot (5) \cdot (3) \cdot (1) \end{aligned}

15!! = 1 \cdot 3 \cdot 5 \cdot 7 \cdot 9 \cdot 11 \cdot 13 \cdot 15 = 2027025

Combined equation that will handle both even and odd numbers:

\begin{aligned} N!! = \prod_{n = 0}^{⌈ \frac{N}{2} ⌉ - 1} (N - 2 n) = N \cdot (N - 2) \cdot (N - 4) \cdot (N - 6) \cdot . . . \end{aligned}

This is the equation implemented by the function.

Below is a table of double factorials for N = −1 to N = 100, followed by the PHP source code used to compute it. The table was computed using the above double factorial function.

Double Factorial Table for N = −1 to 100

  N     N!!
 -1!! = 1
  0!! = 1
  1!! = 2
  2!! = 3
  3!! = 8
  4!! = 15
  5!! = 48
  6!! = 105
  7!! = 384
  8!! = 945
  9!! = 3840
 10!! = 10395
 11!! = 46080
 12!! = 135135
 13!! = 645120
 14!! = 2027025
 15!! = 10321920
 16!! = 34459425
 17!! = 185794560
 18!! = 654729075
 19!! = 3715891200
 20!! = 13749310575
 21!! = 81749606400
 22!! = 316234143225
 23!! = 1961990553600
 24!! = 7905853580625
 25!! = 51011754393600
 26!! = 213458046676875
 27!! = 1428329123020800
 28!! = 6190283353629375
 29!! = 42849873690624000
 30!! = 191898783962510625
 31!! = 1371195958099968000
 32!! = 6332659870762850625
 33!! = 46620662575398912000
 34!! = 221643095476699771875
 35!! = 1678343852714360832000
 36!! = 8200794532637891559375
 37!! = 63777066403145711616000
 38!! = 319830986772877770815625
 39!! = 2551082656125828464640000
 40!! = 13113070457687988603440625
 41!! = 107145471557284795514880000
 42!! = 563862029680583509947946875
 43!! = 4714400748520531002654720000
 44!! = 25373791335626257947657609375
 45!! = 216862434431944426122117120000
 46!! = 1192568192774434123539907640625
 47!! = 10409396852733332453861621760000
 48!! = 58435841445947272053455474390625
 49!! = 520469842636666622693081088000000
 50!! = 2980227913743310874726229193921875
 51!! = 27064431817106664380040216576000000
 52!! = 157952079428395476360490147277859375
 53!! = 1461479318123759876522171695104000000
 54!! = 8687364368561751199826958100282265625
 55!! = 81842841814930553085241614925824000000
 56!! = 495179769008019818390136611716089140625
 57!! = 4746884825265972078944013665697792000000
 58!! = 29215606371473169285018060091249259296875
 59!! = 284813089515958324736640819941867520000000
 60!! = 1782151988659863326386101665566204817109375
 61!! = 17658411549989416133671730836395786240000000
 62!! = 112275575285571389562324404930670903477890625
 63!! = 1130138339199322632554990773529330319360000000
 64!! = 7297912393562140321551086320493608726062890625
 65!! = 74589130387155293748629391052935801077760000000
 66!! = 488960130368663401543922783473071784646213671875
 67!! = 5072060866326559974906798591599634473287680000000
 68!! = 33738248995437774706530672059641953140588743359375
 69!! = 355044260642859198243475901411974413130137600000000
 70!! = 2395415678676082004163677716234578672981800778515625
 71!! = 25563186766285862273530264901662157745369907200000000
 72!! = 174865344543353986303948473285124243127671456831640625
 73!! = 1891675820705153808241239602722999673157373132800000000
 74!! = 13114900840751548972796135496384318234575359262373046875
 75!! = 143767362373591689426334209806947975159960358092800000000
 76!! = 1009847364737869270905302433221592504062302663202724609375
 77!! = 11213854265140151775254068364941942062476907931238400000000
 78!! = 79777941814291672401518892224505807820921910393015244140625
 79!! = 897108341211212142020325469195355364998152634499072000000000
 80!! = 6462013286957625464523030270184970433494674741834234775390625
 81!! = 73562883979319395645666688474019139929848516028923904000000000
 82!! = 536347102817482913555411512425352545980058003572241486357421875
 83!! = 6179282254262829234236001831817607754107275346429607936000000000
 84!! = 45589503739486047652209978556154966408304930303640526340380859375
 85!! = 531418273866603314144296157536314266853225679792946282496000000000
 86!! = 3966286825335286145742268134385482077522528936416725791613134765625
 87!! = 46764808100261091644698061863195655483083859821779272859648000000000
 88!! = 352999527454840466971061863960307904899505075341088595453568994140625
 89!! = 4208832729023498248022825567687608993477547383960134557368320000000000
 90!! = 32122956998390482494366629620388019345854961856039062186274778466796875
 91!! = 387212611070161838818099952227260027399934359324332379277885440000000000
 92!! = 2987435000850314871976096554696085799164511452611632783323554397412109375
 93!! = 36397985440595212848901395509362442575593829776487243652121231360000000000
 94!! = 283806325080779912837729172696128150920628587998105114415737667754150390625
 95!! = 3494206602297140433494533968898794487257007658542775390603638210560000000000
 96!! = 27529213532835651545259729751524430639300973035816196098326553772152587890625
 97!! = 342432247025119762482464328952081859751186750537191988279156544634880000000000
 98!! = 2725392139750729502980713245400918633290796330545803413734328823443106201171875
 99!! = 34243224702511976248246432895208185975118675053719198827915654463488000000000000
100!! = 275264606114823679801052037785492781962370429385126144787167211167753726318359375

Here is the complete PHP program source code used to generate the above table.

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// Double factorial table for n = 1 to 100

   $EndOfTable = 100;

   $TableStr = '';

   for ($n = -1;   $n <= $EndOfTable;   $n++)
{
   $TableStr .= sprintf("% 3d", $n)."!! = ".bcDouble_Fact($n)."\n";
}
   print "<pre>bcDouble Factorial Table\n\n  N     N!! \n$TableStr</pre>";


   function bcDouble_Fact ($IntN)
{
   $N = trim($IntN);   if (!is_numeric($N) or $N < -1) {return FALSE;}
   $j = ceil($N/2) - 1;
   $P = 1;  for($n=0;  $n <= $j;  $n++) {$P = bcmul($P, $N - 2*$n);}

   return $P;
}

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Ref:

http://mathworld.wolfram.com/DoubleFactorial.html

Jay Tanner - 2024