How do I evaluate this summation?

Question

I was wondering how do I solve the summation? $$\frac{1}{n}\sum_{j=1}^nja_{m,j-1}a_{m-1,j-1}$$ I found it in this link on power towers

http://mathworld.wolfram.com/PowerTower.html

Answer 1

The actual definition is

$$a_{m,n}= \begin{cases} 1 & \text{if }n=0\\ \frac{1}{n!}& \text{if }m=1 \\ \frac{1}{n}\sum_{j=1}^n j a_{m,n-j} a_{m-1,j-1} & \text{otherwise} \end{cases} $$

So to construct this array you can think of $a_{m,n}$ as a function $f(m,n)$ defined recursively as

$$f(m,n)= \begin{cases} 1 & \text{if }n=0\\ \frac{1}{n!}& \text{if }m=1 \\ \frac{1}{n}\sum_{j=1}^n j\times f(m,n-j)\times f(m-1,j-1) & \text{otherwise} \end{cases} $$ with $m,n\in\mathbb{N}$. Evaluating the summation simply requires you to find the values of $f(m,n)$ at $(m,n-j)$ and $(m-1,j-1)$ for every value of $j$.