f(x) = exp(alteration of f's own Taylor-series)

[Leroy Quet]

I sent this to sci.math, but no one has yet replied there.
If I figured out the sequence below correctly, it is not in the EIS now.

Leroy
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Let f(x) be an analytic function (about x=0).
Let f_k = k_th derivative of f(x) at x=0.

(So, f(x) = sum{k=0 to oo} (x^k/k!) f_k .)


So, for n = a fixed integer >= 2, 

What is f(x), if f(x)  = 


exp(sum{k=0 to oo} (x^(n^k)/(n^k)!) f_k ) ?    

 
Now,
f_0 = 1.

f_{j+1} =

sum{k=0 to floor(ln(j+1)/ln(n))} binomial(j,n^k -1) f_{j+1-n^k) f_k,

I think.


If n=2, say, I get (by-hand and carelessly, so may be wrong):

f_k -> 1, 1, 2, 3, 10, 26,...

Is there a closed form for this sequence, or for any particular n >= 2?


thanks,
Leroy Quet