# The following sequence has been encrypted:

1-0

3-0

6-1

126-2

373378-2

20532303792-2

24311068187968-3

Can you find the pattern and the next 3 elements?

## Hint a - 2nd April 2020

I'm here giving you the next 2 elements. The pattern and the third element remain for your :)

7356612131551389558912-4

651915846947455861534362510912-5

## Hint b - 10th April 2020

Factorials are exceedingly used

## Hint c - 15th April 2020

I really thought this puzzle will be solved quickly and without hints ^^, it's among the easiest I posted here! Alright, giving you another hint, but it is becoming like... so easy!

At step $$n$$, there are $$n+1$$ factorials used!

## Hind d - 3rd May 2020

I'm removing the encrypted part of this puzzle with this hint!

The sequence's numbers are:
1
3
60
12600
37337800
2053230379200
24311068187968000

And so on :)

# Hint e - 6th May 2020

Giving you an equivalent function ;)

## Defining $$H(n)$$

This puzzle is about $$n+1$$ factorials so I told myself, why not use Stirling's formula?
$$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$

Suppose that $$P(n)$$ is the sequence we want to understand. $$P$$ for Puzzle
$$P(1) = 1$$
$$P(2) = 3$$
$$P(3) = 60$$ and so on.
I'm here going to define $$H(n)$$, $$H$$ for Hint, using Stirling's formula and such as $$P(n)\sim H(n)$$ where the sign $$\sim$$ means that the two quantities are asymptotic: their ratio tends to 1 as n tends to infinity.

Let $$T_n = \dfrac{n(n+1)}2$$ and $$\displaystyle K(n) = \prod_{k=1}^{n}k^k$$ ($$K$$ is the hyperfactorials) I computed for you: $$H(n) = \dfrac{\pi^{\frac{1-n}2}\sqrt{(n+1)}T_n^{T_n}}{\sqrt{(n-1)!}K(n)}$$

## Computing $$H(n)$$

The following Julia script gave me the first elements of $$H(n)$$

|+--------------------+-------------------------------+
| P(n) | H(n) rounded at one decimal |
|+--------------------+-------------------------------+
| 1 | 1 |
| 3 | 3.3 |
| 60 | 68.8 |
| 12600 | 14823.8 |
| 37337800 | 45388978.9 |
| 2053230379200 | 2501368479610.3 |
| 24311068187968000 | 3000139818683595001.3 |
|+--------------------+-------------------------------+

 function T(n)
$$~~~~$$ n*(n+1)/2
 end

 function K(n)
$$~~~~$$ k = 1
$$~~~~$$ for i in 1:n
$$~~~~~~~~$$ k = k*i^i
$$~~~~$$ end
$$~~~~$$ k
 end

 function Hint(n)
$$~~~~$$ pi^((1-n)/2)*sqrt(n+1)*(T(n)^T(n))/(sqrt(factorial(n-1))*K(n)sqrt(2)^n)
 end

## Plots

And I even made some plots for you :)

## Warning!

The fact that the definition of $$H(n)$$ is really (really) complicated doesn't implies $$P(n)$$ is also complicated! It's even the opposite, $$P(n)$$ is simple to express. Think of it like the Stirling's formula, $$n!$$ is really simple to express and $$\sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$ is a little less simple to write.

# Note

Please note that I've changed the values of my sequence after Hint d to make it even easier :)

• Are you sure that the sequence is encrypted? It seems like maybe something involving prime numbers but not cryptography... Mar 27, 2020 at 20:03
• @dan1st, yes, it is. I have encrypted it because, otherwise, it would be too easy finding the pattern :) You made me realize, I forgot the cryptography tag! Many thanks ;)
– JKHA
Mar 27, 2020 at 20:47
• Heh, I thought your avatar was a piece of the Mandelbrot set until I clicked through to your profile :-) Mar 27, 2020 at 20:52
• @Randal'Thor, haha! You did find one of the reason why I chose it :)
– JKHA
Mar 27, 2020 at 20:59
• @Randal'Thor, By the way, I went for yours and the link in your description: riddled.azurewebsites.net isn't working anymore
– JKHA
Mar 28, 2020 at 0:54

$${{1+2+\cdots+n}\choose{1;2;\dots;n}}=\frac{(1+2+\cdots+n)!}{1!\,2!\,\dots\,n!}$$