Find an explicit formula for the sequence whose first few terms are 1/3, 2/3, 2, 8, 40, 240...

Question

I am trying to find the formula for this sequence:

$$1/3, 2/3, 2, 8, 40, 240...$$

I know I need to use a factorial in the formula, and then raise it to a power of something maybe? But from there I'm lost on what will work. I tried $n^3$, and a ton of other ones but nothing is working.

Answer 1

First off, take out a factor of $\frac{1}{3}$. The sequence is then: $$1, 2, 6, 24, 120, 720, ...$$ Which is just $u_n=n!$.

So the required formula is: $$u_n=\frac{n!}{3}$$

Hope this helps!

Answer 2

This is clearly $\frac{N!}{3}$. You may like to go through A002301.