Q:

Trinomial Pascal's Triangle

I know that there's a trinomial theorem (and a multinomial theorem), but I was wondering if there was a similar structure for trinomials as there is for binomials, like Pascal's triangle. 
Thanks in advance! 

A:

Pascal's pyramid is a series of triangles that tell us the coefficients of terms in powers of trinomials.  If the triangles are stacked in succession from top to bottom, we get a pyramid, so it's called Pascal's pyramid.  This can be really interesting to explore, so I suggest that you try to calculate the first few triangles. 
The way it works is that you start with $1$ around all three corners and then, on the inside, you add up the numbers from the previous triangle to get your new term.  The first triangle is:
$$1$$
Then we have:
$$1 \\ 1 \ \ \  1$$
Now, the third row is when the new terms come in.  This is a three-sided triangle, so we have two middle terms.  These middle terms are below the $1$ and $1$ on the edge of the previous triangle, so we get $1+1=2$.
$$1 \\ 2 \ \ 2 \\ 1 \ \ 2 \ \ 1$$
Now, the next is a four-sided triangle, so in the middle of the edges, we have two elements.  Each of these are below a $1$ and a $2$, so we get $1+2=3$. Also, we have a middle that is below a $2$, a $2$, and a $2$, giving us $2+2+2=6$:
$$1 \\ 3 \ \ 3 \\ 3 \ \ 6 \ \ 3 \\ 1 \ \ 3 \ \ 3 \ \ 1$$
From then on, it gets more complicated, but hopefully, you get the point.  Can you figure out a formula to connect the elements in each triangle to the trinomial coefficient $\frac{n!}{x!y!x!}$?  Can you figure out how to connect each element to the coefficient of a term in the power of a trinomial?  Good luck!