How many paths are there from one vertex of a subdivided triangle to another? Other than counting, is there any shortcut to solve this?
Using only the paths and the indicated directions, how many
different routes are there from $A$ to $J$?
My answer is $22$.

 A: Put a number on every node, starting with a $1$ on point $A$. The numbers you put on node is how many ways to get to this particular node. It is the sum of every nodes that lead to it.
        1
      2   1
    6   4   1
 22  16   6   1

You are right, there are $22$ ways to get from $A$ to $J$.
A: Building upon a prior comment answer, we have:
$$\sum\limits_{k=0}^n C_k {n+k \choose 2k} = \, _2F_1(-n,n+1;2;-1)$$
where $C_n$ is the $n$th Catalan number and $F$ is the HyperGeometric function.
Here's a table:
$$\left(
\begin{array}{cc}
 0 & 1 \\
 1 & 2 \\
 2 & 6 \\
 3 & 22 \\
 4 & 90 \\
 5 & 394 \\
 6 & 1806 \\
 7 & 8558 \\
 8 & 41586 \\
\end{array}
\right)$$
A: The numbers given by @David G. Stork in comments and answers are the Schröder numbers $S_n$ 
Quoting the Wikipedia page they
"describe the number of lattice paths from the southwest corner $(0,0)$ of an $(n \times n)$ grid to the northeast corner $(n,n)$ using only single steps north, $(0,1)$; northeast, $(1,1)$ or east, $(1,0)$ that do not rise above the $\text{SW-NE}$ diagonal."
You could find them in sequence $A006318$ in $OEIS$.
