# Did I find out something in my dreams? (I)

One day, I was asleep, and in my dream yesterday, I dreamt that I was playing with factorials. I then got this sudden revelation of a weird formula:

$$\LARGE{\frac{1! × 2! × ... × n!}{1 × 2 × ... × 2n}}$$

Simplifying this, we can get:

$\LARGE{\frac{G(n + 2)}{(2n)!}}$

Where $G(z)$ is the $\text{Barnes G-function}$ and $G(n + 2)$ is the superfactorial

When we substitute $n$ with different values, we gain these numbers:

 $n = 3$ $0.01666…$ $n = 4$ $0.00714285…$ $n = 5$ $0.00952380…$ $n = 6$ $0.0519480…$ $n = 7$ $1.438561…$

I decided to plot this using WolframAlpha, but I do not have the Pro Version, so I plotted a smaller range of values instead (because I was bored a lot...)

Does this graph or number set have any special value to them? I was just bored, but maybe you can find a better explanation for these numbers. I would love to see what random facts you can gain about these weird numbers that came to me in a dream...

Note by A Former Brilliant Member
1 week, 5 days ago

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Do these numbers look like something similar, like any other number?

Please give in a small research and let me know...

- 1 week, 5 days ago

@Hamza Anushath, Wait! I got it! I know what's special about these numbers! I figured it out!

They all are.....

Reply to this comment to know the answer.

- 1 week, 5 days ago

Decimal numbers (Irrational I think, but could be rational)

Wow! This is my 2nd biggest discovery after I discovered that 1+1 = 2!

LOL, XD!!! LOL, XD!!! LOL, XD!!! LOL, XD!!!

- 1 week, 5 days ago

Yes @Yashvardhan Pattanashetti

Plus, they are rational numbers, as they repeat their digits forever

- 1 week, 5 days ago

awesome

- 1 week, 5 days ago

Maybe you should have written:

I know how to solve it, but this comment section is too small for my proof.

Yours sincerely, Fermat.

- 1 week, 5 days ago

Nice one, @Vinayak Srivastava!

- 1 week, 5 days ago

LOL!! nice!!

- 1 week, 5 days ago

Maybe replace the Fermat with @Yashvardhan Pattanashetti's name?

What do you think?

- 1 week, 5 days ago

I'd rather have it be Fermat

- 1 week, 5 days ago

Ok, then. But good joke, no?

- 1 week, 5 days ago

Yup! It was awesome!!

- 1 week, 5 days ago

Ok, then.

- 1 week, 5 days ago

I guessed you made a discovery in my dreams. I was still awake around $10:30$ when I suddenly thought about new problem series:

Binary clocks

Algebraic binary locks

Basically, I am a walking-talking mathematician as well. @Hamza Anushath, @Yashvardhan Pattanashetti, @Páll Márton

- 1 week, 5 days ago

Woah, are you a walking... talking... mathematician!?!

I am just a minor mathematician...

- 1 week, 4 days ago

In my sleep, I am talking about being a walking-talking mathematician.

- 1 week, 4 days ago

Do you think I am a walking-talking mathematician?

- 1 week, 4 days ago

Yes

- 1 week, 4 days ago

Ok.

- 1 week, 4 days ago

Not the zombie type of walking-talking, though.

- 1 week, 5 days ago

We get that. lol

- 1 week, 5 days ago

Ok. But what do you think?

Do you think I am a walking-talking mathematician?

- 1 week, 5 days ago

Talking-yes, walking, i havent seen you walk, mathematician yes

- 1 week, 5 days ago

Ok, then.

What do you think about my new problem series that I mentioned?

- 1 week, 5 days ago

I think its a great idea!!

- 1 week, 5 days ago

Just thought of two more problem series:

Binary locks

What do you think, @Yashvardhan Pattanashetti?

- 1 week, 5 days ago

Nice!! Looking forward to it.

- 1 week, 5 days ago

Posting the first hexadecimal lock problem in $5$ mins.

- 1 week, 5 days ago

Ok, good luck

- 1 week, 5 days ago

Wait - change that to binary locks - hexadecimal locks will never work.

- 1 week, 5 days ago

Posting it...

- 1 week, 5 days ago

... now

- 1 week, 5 days ago

The only thing I was able to link this to was the superfactorial, defined by Neil Sloane and Simon Plouffe to be the product of incrementing factorials (the numerator in your expression). Using the notation for a superfactorial, your expression could simplify to:

$\dfrac{sf(n)}{(2n)!}$

- 1 week, 5 days ago

Thanks a lot @David Stiff! If you could, could you tell me whether the graph I generated is correct or not...?

- 1 week, 5 days ago

You're welcome Hamza!

I wrote some Python code to generate a graph of this expression, and both the numerical values and the graph are identical to those you calculated. I only used the superfactorial form however, not the more general Barnes G-function. I don't think I can post pictures in a reply, but here are the first 10 values I got:

$1, 0.5, 0.08\overline{3}, 0.01\overline{6}, 0.007142857142857143, 0.009523809523809525, 0.05194805194805195, 1.4385614385614385, 241.67832167832168$

I find it interesting that the graph first descends, bottoms out at $n = 4$ and then ascends again, rocketing up at $n = 6$.

- 1 week, 4 days ago

Thanks a lot once again @David Stiff

P.S. We can post pictures in a comment by uploading it in a note and copying that code and pasting it here, the code starting with ![]

- 1 week, 4 days ago

No problem. And thanks for the tip!

- 1 week, 3 days ago

You can't generate a graph using this function, @David Stiff, @Hamza Anushath

- 1 week, 4 days ago

I was just thinking, I could simplify your top equation:

$\frac{n!!}{2n!}$

Try it, @Hamza Anushath

- 1 week, 4 days ago

$n!!$ is the double factorial. @Hamza Anushath

- 1 week, 4 days ago

@Yajat Shamji, it's not the double Factorial, it is actually a superfactorial...

- 1 week, 4 days ago

Oops. But did you try it?

- 1 week, 4 days ago

Yes, and very sorry, it brought a wrong value @Yajat Shamji

- 1 week, 4 days ago

Well, at least you tried.

- 1 week, 3 days ago