calculation and analysis: evaluate the higher order derivative at some point under some assumptions

I need to evaluate the $ k ^ { rm th} $ derivative of order

re[w

which is really equal to

Sum[Binomial[k, j] re[w

to t = 0. I get that w[0]== 0 Y w & # 39;[0]== 0.

I'm trying

Evaluate[RE[D[RE[D[w

and so

FullSimplify[3w'[3w'[3w'[3w'[0]w & # 39; & # 39;[0] + w[0]w & # 39; & # 39; & # 39;[0], Assumptions -> w[0] == 0 && w & # 39;[0] == 0]

but this works only for fixed k.

Is there any way to do the evaluation in t = 0 under the assumptions w[0]== 0 Y w & # 39;[0]== 0 for arbitrary k and also take into account the value of evaluations up to k-1 order? Any help is appreciated. Thank you.