I am stuck while trying to solve a recurrence relation. I just need the characteristic function to know the dependence of $Y_n$.
My recurrence relation is of the following form where $a$ and $b$ are constant,
$Y_n = a + b \frac{\sum_{j=1}^{n-1}Y_j}{n-1}$
I am unable to figure out the characteristic equation or even the asymptotic nature of the series.
Using the definition of $X_n$ in the comments by @Paul Sinclair, I ran out the first couple of cases, and then used the Online Encyclopedia of Integer Sequences to formulate the following conjecture: $$X_n = \frac{1}{(n-1)!}\left[\displaystyle\sum_{i=0}^{n-2} \left(ab^i\sum_{j=i+2}^{n} s(n,j)\right)+X_1\prod_{i=1}^{n-1}(b+i)\right]$$ where $s(n,k)$ is the unsigned Stirling number of the first kind.
Using the recurrence relation $X_{n+1}=a+\frac{n+b}{n}X_n$ given by Paul Sinclair, it is an interesting exercise to prove the formula using induction.