How to get a simple recurrence relation from $v_n = n - 1 + \frac{2}{n} \sum_{k=1}^{n-1} v_k$?

Question

I've got this sequence :

$v_1 = 0, v_2 = 1$ that verifies: $v_n = n - 1 + \frac{2}{n} \sum_{k=1}^{n-1} v_k$

And I'm trying to get a simple recurrence relation between $v_{n+1}$ and $v_n$?

I've tried to substract $v_n$ from $v_{n+1}$, and to look for some kind of expression $u_n = f(v_n)$ to make it easier but so far all of my attempts have failed.

Answer 1

We have $$nv_n-(n-1)v_{n-1}\\ =n(n-1)-(n-1)(n-2)+2v_{n-1}$$ Hence $$v_n={n+1\over n}v_{n-1}+2{n-1\over n},\ n\ge 3$$ It is possible to solve the relation explicitly, as $${v_n\over n+1}={v_{n-1}\over n} +2{n-1\over n(n+1)}={v_{n-1}\over n}+{4\over n+1}-{2\over n}$$ Thus $${v_n\over n+1}=4\sum_{k=1}^n{1\over k+1}-2\sum_{k=1}^n{1\over k},\quad n\ge 1$$