Right now I am attempting to solve the recurrence thats solves a famous problem (the colored balls problem: https://mathoverflow.net/questions/41939/a-balls-and-colours-problem) given by
$2i(n-i)Z_i = n(n-1) + (n-i)(i+1)Z_{i+1} + (n-i)(i-1)Z_{i-1}$
with the base case $Z_n=0$. Apparently $Z_1 = (n-1)^2$, which is what I want to show in order to solve the problem. So far I have tried to do this telescoping approach where I manipulate the above recurrence to yield
$(iZ_i - (i+1)Z_{i+1}) - ((i-1)Z_{i-1} - iZ_i)=\frac{n(n-1)}{n-i}$
from which summing from $i=1$ to $n-1$ we can determine then that
$(n-1)Z_{n-1} - Z_1 = \sum\limits_{i=1}^{n-1} \frac{n(n-1)}{n-i}$
but it seems impossible to get rid of the $Z_{n-1}$ term. Any suggestions? Thanks!