Closed-form solution to a recurrence of all previous terms

Question

I am looking for a method/formula to solve recurrence relations of this form: $$ U_{n+1}= \sum_{r=0}^{n} a_{r}U_{r} \ with \ U_0=x $$ Where the coefficients are in terms of r. This is the only case I could work out:

$$ U_{n+1}= \sum_{r=0}^{n} U_r \ with \ U_0 =x \implies U_n = 2^nx$$

Any suggestions would be great (even saying that this is not worth the effort).

Answer 1

Writing out the first few terms, it seems there's a product relation for the $U$'s in terms of the $a$'s: $$U_{n+1}=a_0x\prod_{i=1}^n(1+a_i) .$$ I'm sure a quick induction will confirm this.