Trying to figure out the pattern in this series

Question

I'm trying to figure out the rule that's producing the series below. The first column is the factorials $1!,2!,3!,4!,5!,6!$ But I can't figure out what else is going on. So I know the next row will start with $7!=5040$, but can't say much beyond that. If someone has insight as to what the next numbers in the series are that would be much appreciated.

$1$

$2\qquad2$

$6\qquad12\qquad6$

$24\qquad72\qquad72\qquad24$

$120\qquad480\qquad720\qquad480\qquad120$

$720\qquad3600\qquad7200\qquad7200\qquad3600\qquad720$

Thanks

Answer 1

The easiest way to find the answer here is to divide out by the first terms, producing:

$$\begin{array} &&&&&&1\\ &&&&1&&1\\ &&&1&&2&&1\\ &&1&&3&&3&&1\\ &1&&4&&6&&4&&1\\ \end{array}$$ ...which should look awfully familiar. This implies that the $(n,k)$ entry of your series is $n! {n-1 \choose k}$.