<s>[INST] recurrence relation with variable coefficients How to solve recurrence relation $y_n= y_{n-1}+ (n-1).y_{n-2}$ where $n$ is a variable ?
$y_n$ is a $n$th term, $y_{n-1}$ is $(n-1)$th term and $y_{n-2}$ is $(n-2)$th term.
 [/INST] Hint
Let $y_{0}=a,y_{1}=b$,and let
$$f(x)=\sum_{n=0}^{\infty}y_{n}x^n$$
then we have
$$y_{n}x^n=xy_{n-1}x^{n-1}+x^2(n-2)y_{n-2}x^{n-2}+y_{n-2}x^n$$
$$\Longrightarrow \sum_{n=2}^{\infty}y_{n}x^n=x\sum_{n=2}^{\infty}y_{n-1}x^{n-1}+x^2\sum_{n=2}^{\infty}(n-2)y_{n-2}x^{n-2}+x^2\sum_{n=2}^{\infty}y_{n-2}x^{n-2}$$
$$\Longrightarrow f(x)-a-bx=x[f(x)-a]+x^3f'(x)+x^2f(x)$$
so
$$f'(x)+\dfrac{x^2+x-1}{x^3}f(x)=\dfrac{(a-b)x-a}{x^3}$$
so
$$f(x)=e^{-\int\frac{x^2+x-1}{x^3}dx}\left(\int\dfrac{(a-b)x-a}{x^3}e^{\int\frac{x^2+x-1}{x^3}dx}dx+C\right)$$
then I think you can do it,even it's ugly
</s>