Triangle of Multinomial Coefficients

Question

What is the "Triangle Of Multinomial Coefficients" seen here: http://oeis.org/A036038 (OEIS: A036038)

I can see that the diagonals of this triangle are just factorials... for example the last number in each row is as follows: 1,2,6,24,120 and that's just 1!, 2!, 3!, 4!, 5! And this process repeats for the second to last number except instead of using factorials it's 1*2*3*4*5 where the initial number isn't 1 but is instead something else (so factorial times a constant) and that constant is the number from the previous diagonal.

Is that all there is to it?

Answer 1

Each term in the $n$th row is the number of ways to order the corresponging unordered partition of $n$. The terms in each row are ordered with partitions into the fewest parts first, and partitions with the same number of parts are ordered by largest part, then by second largest part etc.

e.g. In the $4^{th}$ row,

$\begin{array}\\ 1 & 4&6&12&24\\ 4&3+1&2+2&2+1+1&1+1+1+1\\ \frac{4!}{4!}&\frac{4!}{3!.1!}&\frac{4!}{2!.2!}&\frac{4!}{2!.1!.1!}&\frac{4!}{1!.1!.1!.1!} \end{array}$