How do I solve a general formula for a very complicated list of numbers

Question

So I have been given a table of numbers and I have figured out the pattern but have no idea what the general formula for the sequence is. Here is the sequence:

n	m
2	8
3	30
4	128
5	650
6	3912

I realised that the m-value is equal to the n-value multiplied by the previous m-value and then 2*n-value is added to create the m-value, but I'm not sure how to put this into a formula. Please help it would be much appreciated, and explanations as to how to do these in the future are much appreciated.

Answer 1

HINT:

Although there can be infinitely many solutions, the way you can write your relation is in the form of a recurrence relation. You can think of the $m$ value that corresponds to $n$ as $f(n)$, and the previous $m$ value as $f(n-1)$.

With these, try to define $f(n)$ in terms of $f(n-1)$ and $n$ according to your pattern.

Answer 2

Ok so I asked a very smart friend and he came up with this more simplified version than JMoravitz in the comments. Its the exact same but easier on the eyes ;)

$f(n)$ $=n(f(n-1))+2n$

Thanks everybody for all the help!

Answer 3

The recurrence sequence defined by $f(n) = n f(n-1)+2n$ with $f(2)=8$ can be explicitly calculated as $$f(n) = 2 \sum_{k=0}^{n-1}\frac{n!}{k!}$$