The equation $f(f(x))=xe^x=x+x^2+\frac{x^3}{2}+\frac{x^4}{6}+\dots$ has a unique formal powerseries solution. Is its convergence radius 0 as was shown by Baker for the equation $f(f(x))=e^x1$? Or more generally: Let $F(x)=xe^x$ does the unique formal powerseries solution of $F(f_t(x))=f_t(F(x))$ with $f_t(x)=x+t x^2 +\dots$ have convergence radius 0 for all noninteger numbers $t$?
I don't have an analytically founded answer, but like in the thread on the iteration of sin(x)
formal power series convergence
I've plotted the characteristics of the coefficients of the powerseries for the halfiterate of x*exp(x).
In the first plot we see, that the log of the absolute values of the coefficients grow nearly linearly with their index, so this is a hypergeometric series. See growth. 

