Search for a tool
Double Factorial

Tool for calculating the double factorial. The double factorial n!! is the product of non-zero positive integers less than or equal to n that have the same parity as n (even or odd).

Results

Double Factorial -

Tag(s) : Arithmetics

Share
Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!


Please, check our community Discord for help requests!


Thanks to your feedback and relevant comments, dCode has developped the best 'Double Factorial' tool, so feel free to write! Thank you !

Double Factorial

Double Factorial Calculator N!!


Tool for calculating the double factorial. The double factorial n!! is the product of non-zero positive integers less than or equal to n that have the same parity as n (even or odd).

Answers to Questions

How to calculate a double factorial?

The double factorial (also called semifactorial) of a number $ n $, denoted by $ n!! $ (with 2 exclamation marks), is the multiplication (the product) of all non-zero integers less than or equal to $ n $ that have the same parity.

$$ n!! = \prod_{k=0}^{\left\lceil\frac{n}{2}\right\rceil - 1} (n-2k) = n (n-2) (n-4) \cdots $$

If $ n $ is an even number (multiple of 2) then $ n !! $ is the multiplication of all multiples of $ 2 $ less or equal than $ n $ (and greater than $ 0 $)

If $ n $ is an odd number (not a multiple of 2) then $ n !! $ is the multiplication of all non-multiple numbers of $ 2 $ less or equal than $ n $ (and greater than $ 0 $).

Example: $$ 8!! = 2 \times 4 \times 6 \times 8 = 384 $$

Example: $$ 5!! = 1 \times 3 \times 5 = 15 $$

Be careful not to confuse the double factorial $ n !! $ with the factorial of factorial $ (n!)! $

What is the value of double factorial of 0?

By convention, the double factorial of zero is equal to 1: $ 0 !! = 1 $

What are the first values of the double factorial function?

The values of the first double factorials: $$ 0!! = 1 \\ 1!! = 1 \\ 2!! = 2 \\ 3!! = 3 \\ 4!! = 8 \\ 5!! = 15 \\ 6!! = 48 \\ 7!! = 105 \\ 8!! = 384 \\ 9!! = 945 \\ 10!! = 3840 $$

What are the properties of double factorial?

The remarkable relations of the double factorial with the factorial are:

$$ n! = n!! (n-1)!! $$

$$ n!! = \frac{n!}{(n-1)!!} = \frac{(n+1)!}{(n+1)!!} $$

Source code

dCode retains ownership of the online 'Double Factorial' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Double Factorial download for offline use on PC, tablet, iPhone or Android !

Need Help ?

Please, check our community Discord for help requests!

Questions / Comments

Thanks to your feedback and relevant comments, dCode has developped the best 'Double Factorial' tool, so feel free to write! Thank you !


Source : https://www.dcode.fr/double-factorial
© 2020 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.
Feedback