Fun beginner-level algebra riddle

[Yoni]

Find a closed-form(*) expression for: 1 * 3 * 5 * ... * (2n - 1).



(*) What is "closed-form"?
This is a mathematical term that is not really well-defined.
Mathematicians say "closed-form" to mean "a nice-looking expression(**) that doesn't have ugly stuff like "..." in it".
For example, an acceptable closed-form expression for 1 * 2 * 3 * ... * n would be n!.

(**) That is, an expression with no unresolved sums, products, integrals and functions which are not included in the unofficial set of "Functions That Count As Closed-Form Anyway".

[Rule]

Hmmm...  see the black text below.  Would this count?  It might qualify as an "unresolved product," but I'm not very good with jargon, so maybe it's what you want.


(Capital PI) from i = 1 to i =n of (2i-1).  Basically, the product of all odd numbers.

[shout]

Sum(Xn) = ((2n)! + -1)

I don't know if that would work. I'm too lazy at the moment to see if that would actully work.

[Yoni]

Rule: No, that doesn't count as closed-form - it has an unresolved product, like you said.
What I want is an answer more like Shout's, except he's wrong, so keep trying.

[shout]

n
∑ = (2n + -1)
n = 1

Hmmm? Don't laugh at my attempt at sigma notation!

[Yoni]

That is not only not closed-form, but even when treated as open-form, wrong.

[111787]

I know this...

((2^(n-1))((2n-1)!))/(((2)^(n-1))((n)^(n-1)))

It works.

[R.a.B.B.i.T]

Find a closed-form expression for: 1 * 3 * 5 * ... * (2n - 1).

t_n = t_{n - 1} * (2n - 1) , n =/= 1
probably wrong, but worth a shot :\

[Yoni]

111787: Congratulations on the first serious attempt so far. People keep interpreting "closed-form" as "not closed-form". This is an "algebra" riddle, not a "mathematical notation" riddle.

Wrong. Try again!

Ex. 1*3*5 = 15.
Your expression with n=3 (2*3 - 1 = 5) gives 40/3 which is not 15.

[111787]

Its so hard to make it defined for 1.

Next attempt:
((2n-1)!)/((2n-1)^n)

[dxoigmn]

I think I got it.
(2n-1)!/(2^(n-1)*(n-1)!)

[Yoni]

dxoigmn solved it. Good job!
I know an equivalent expression that looks slightly better than this. See the black text below.
Everyone else is left with the task of proving it.


(2n)! / (n! * 2^n)
Can you prove it's equivalent to yours?

[R.a.B.B.i.T]

I still haven't learned factorials or whatever (!).  Hrm...

[dxoigmn]

Can you prove it's equivalent to yours?

Multiple top and bottom of my expression by 2n.

[111787]

dxoigmn is not right i believe:
I can't make dxoigmn's work.
(2n-1)!/(2^(n-1)*(n-1)!)
n = 4
7!/(2^((3*3!))
7!/(2^18)
5040/262,144<> 105

On the other hand
(2n)! / (n! * 2^n)
n = 4
8!/(4!*16)
40320/384 = 105