I have the following iteration $$ y_n\mapsto n\sum_{m=1}^{n+1}y_m $$ starting with $y_1$ being a positive real number. Is there a standard method to find the coeffcients of all $y_n$ after $k<\infty$ iterations? The recursion (?) is applied at every step - here is the first three iterations: $$ y_1\mapsto y_1+y_2\mapsto(y_1+y_2)+2(y_1+y_2+y_3)\\ \mapsto(y_1+y_2)+2(y_1+y_2+y_3)+2((y_1+y_2)+2(y_1+y_2+y_3)+3(y_1+y_2+y_3+y_4)). $$ The parentheses are just for a better orientation. After the second iteration the coefficients are $3,3,2$.
It resembles the iteration method for fractals if $y_1$ was complex; except that here only a finite number of steps interests me ($k\to\infty$ obviously diverges).
Any help appreciated - methods, result, literature.
After computing several iterates and looking for a pattern, it seems that the coefficient of $y_n$ after $k$ iterates of this map ($1\le n\le k+1$) is $$ \frac{(2 k-n+1)!}{2^{k-n+1} (k-n+1)!}. $$ Given this conjecture, it's probably not too hard to prove it by induction.