I was writing a recursive computer algorithm to find the optimal matrix multiplication order (without using memoizing anything, so I am recomputing a lot). Quite interestingly, the triangle below emerged from the number of times each recursive function with different parameters is called. For example, the top of the tree represents the first call: M(1, n). Let M(i, j) be the optimal solution to multiplying Matrix i to j with the least amount of multiplications. Then, in the context of this problem, M(i, j) actually gets its solution by calling M $2 (j-i)$ more, with "smaller" paramters, of course.
Anyways, I don't know if this problem matters much, but we have this cute triangle below. Is there a name for this triangle? Additionally, is it possible to compute an explicit formula for this (I think the recursive definition is $a_{n, m} = a_{n, m-1} + a_{n+1, m-1}$ and $a_{1, 1} = 1$, where $m$ is the height of the node in the triangle below and $n$ is the index of the node counting from the left )?
I would also appreciate any additional mathematical insights into the nature of this, I am not so adept at it.
