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+--- Thread: Double factorial (/thread-3240.html)
Double factorial - salvomic - 03-02-2015 01:49 PM
hi all,
how is a simple (and precise) way to define a command to calculate the "double factorial" (see here) with Prime?
I mean to use it as a simple command.
We cannot write n!!, as that is not the same...
thank you,
Salvo
RE: Double factorial - Han - 03-02-2015 04:18 PM
(03-02-2015 01:49 PM)salvomic Wrote: hi all,
how it a simple (and precise) way to define a command to calculate the "double factorial" (see here) with Prime?
I mean to use it as a simple command.
We cannot write n!!, as that is not the same...
thank you,
Salvo
The wiki explains how you can calculate it using the regular factorial:
If \( n \) is even, then there exists some integer \( k \) such that \( n = 2k \).
\[ n!! = n(n-2)(n-4) \dotsm 4 \cdot 2 = (2k)(2k-2)(2k-4)\dotsm (2\cdot 2) \cdot (2\cdot 1)
= 2^k k! \]
If \( n \) is odd, then there exists some integer \( k \) such that \( n = 2k+1 \). Moreover, \( n-1 \) is even.
\[ n!! =
\frac{\text{product of all integers from 1 to } n}{\text{product of all even integers from 2 to } n-1} =
\frac{n(n-1)(n-2)\dotsm 3 \cdot 2 \cdot 1}{ (n-1)(n-3)(n-5)\dotsm 4 \cdot 2} =
\frac{n!}{(n-1)!!} = \frac{(2k+1)!}{2^k k!} \]
It should be pretty easy to write your own program using these formulas. (Note: the wiki has a slight error with respect to the formula for odd values of \(n \) )
RE: Double factorial - Helge Gabert - 03-02-2015 04:36 PM
No error checking, but something along these lines
#cas
DFA(n):=
BEGIN
IF odd(n) THEN
product(n,n,1,n,2);
ELSE
product(n,n,2,n,2);
END;
END;
#end
Is this what you are asking (simple program)?
RE: Double factorial - salvomic - 03-02-2015 05:15 PM
(03-02-2015 04:18 PM)Han Wrote: The wiki explains how you can calculate it using the regular factorial:
...
It should be pretty easy to write your own program using these formulas. (Note: the wiki has a slight error ...
(03-02-2015 04:36 PM)Helge Gabert Wrote: No error checking, but something along these lines
#cas
...
Is this what you are asking (simple program)?
yes, thanks both!
so this should be enough "precise" and it is ok both in Home and CAS:
Code:
EXPORT factorial(n)
//Double factorial n!!
BEGIN
local f;
IF odd(n) THEN f:=product(X,X,,1,n,2);
ELSE f:=product(X,X,2,n,2); END;
RETURN(f);
END;My memory failed, and I didn't remember "product()"
thanks again,
Salvo
RE: Double factorial - Gerald H - 03-02-2015 05:30 PM
An assist from the 50G side:
::
CK1&Dispatch
BINT1
::
COERCEDUP
BINT2
#/
DUP
FPTR2 ^#FACT
Z2_
ROT
FPTR2 ^PPow#
FPTR2 ^RMULText
SWAP
#0=case
SWAPDROP
SWAP
FPTR2 ^#FACT
FPTR2 ^SWAPRDIV
;
;
RE: Double factorial - salvomic - 03-02-2015 05:32 PM
(03-02-2015 05:30 PM)Gerald H Wrote: An assist from the 50G side:
...
thanks!
50g is always a great Calc!
RE: Double factorial - DrD - 03-03-2015 12:02 PM
Here is a slight variation on your theme:
Code:
EXPORT factorial(n)
//Double factorial n!!
BEGIN
RETURN IFTE(odd(n), product(X,X,1,n,2), product(X,X,2,n,2));
END;-Dale-
RE: Double factorial - salvomic - 03-03-2015 01:24 PM
(03-03-2015 12:02 PM)DrD Wrote: Here is a slight variation on your theme:
...
-Dale-
nice, Dale, thanks!
RE: Double factorial - salvomic - 03-03-2015 06:15 PM
(03-02-2015 05:03 PM)compsystems Wrote: Please HP´ TEAM can include product operator in the next version of ROM, or at least that on the historial, the product command is displayed as an operator...
I agree.
∏, Productoria (or, Italian, Produttoria) as operator, a part of the command product: it would be very useful also.
Tim, do you think that it's possible to do it?
Thank you.
Salvo