Solve recursive equation $ f_n = \frac{2n-1}{n}f_{n-1}-\frac{n-1}{n}f_{n-2} + 1$ Solve recursive equation:
$$ f_n = \frac{2n-1}{n}f_{n-1}-\frac{n-1}{n}f_{n-2} + 1$$
$f_0 = 0, f_1 = 1$
What I have done so far:
$$ f_n = \frac{2n-1}{n}f_{n-1}-\frac{n-1}{n}f_{n-2} + 1- [n=0]$$
I multiplied it by $n$ and I have obtained:
$$ nf_n = (2n-1)f_{n-1}-(n-1)f_{n-2} + n- n[n=0]$$
$$ \sum nf_n x^n = \sum(2n-1)f_{n-1}x^n-\sum (n-1)f_{n-2}x^n + \sum n x^n $$
$$ \sum nf_n x^n = \sum(2n-1)f_{n-1}x^n-\sum (n-1)f_{n-2}x^n + \frac{1}{(1-z)^2} - \frac{1}{1-z} $$
But I do not know what to do with parts with $n$. I suppose that there can be useful derivation or integration, but I am not sure. Any HINTS?

Did you try difference equations? The original expression can be rewritten as (denote $\Delta f_k=f_{k}-f_{k-1}$)
$$
\Delta f_{k}=\frac{k-1}{k}\Delta f_{k-1}+1 =\frac{k-1}{k} \cdot \frac{k-2}{k-1} \Delta f_{k-2} + \frac{k-1}{k}+1= \ldots =\frac{1}{k} \Delta f_1 + \frac{k-1}{k} \\
+ \frac{k-2}{k-1}+\ldots + 1 = \frac{1}{k} \Delta f_1 + k - H_k
$$
By summing both sides over $k$ and using boundary values you should get something like (I haven't checked the algebra!)
$$
f_n= H_n + \frac{n(n+1)}{2}-(n+1)(H_n-1)
$$
