Factorial of $(7/2)!$

Question

It's been many years since I studied maths, and I'm trying to figure out the half factorials $(7/2)!$ without a calculator.

I did $(7/2) \times (5/2) \times (3/2) \times (1/2) = (105/16) ^ \pi = 1.82$ (as per a little tutorial I'm doing).

However, if I put $(7/2)!$ in the calculate I get $11.631$

Am I completely off target here? Sorry if this is elementary - I'm basically starting from the beginning again.

** UPDATE **

Despite the real issue being not knowing how to use the damn calculator properly - I now have a better understanding of this subject thanks to all your comments.

I guess the title has a typo, but in fact I think fractorial is a pretty cool portmanteau! — , Jan 14 '14 at 10:20
$$(7/2)! = \Gamma(7/2+1) = (7/2)\times(5/2)\times(3/2)\times(1/2)\times\sqrt{\pi}$$ while $\Gamma(x)$ is the Gamma function. — achille hui, Jan 14 '14 at 10:22
For half-integer values, there's this Wikipedia article which explains it. — Gigili, Jan 14 '14 at 10:24
(105/16) ^ pi? Google says that's 368.89... Did you mean ^(1/pi)? — arne.b, Jan 14 '14 at 10:26
Perhaps how I'm putting this into the calculator is the issue - maybe I need to learn how to use a scientific calculator :) — Trent, Jan 14 '14 at 10:29
Oops, I ruined your joke by editing the title, sorry @AsalBeagDubh! — Gigili, Jan 14 '14 at 10:30
@arne.b. The answer, as already given by achille hui, is (105 / 16) Sqrt[Pi]. — Claude Leibovici, Jan 14 '14 at 10:38
Both equal signs in (7/2)×(5/2)×(3/2)×(1/2)=(105/16)^π=1.82 are real mysteries to me. — Did, Jan 14 '14 at 10:38
middle equality should be $\frac{105}{16}\sqrt{\pi}$. Got confused perhaps? — user88595, Jan 14 '14 at 10:46

score 2 · Accepted Answer · answered Jan 14 '14 at 11:09

You can work out the half-factorials from the volumes of the spheres, which take the form of

$$ v(n) = \pi^{n/2}d^n/(n/2)! $$

For $n=3$ this gives $4\pi / 3$, makes $(3/2)! = 3 \sqrt{\pi}/4$.

One then finds that $\frac 72! = \frac 72 \frac 52 \frac 32 \frac 12 \sqrt\pi$

Factorial of $(7/2)!$

1 Answers1