I'm having some trouble showing that two things I really want to be the same are in fact the same. I want to show that these two sequences are, in fact, the same thing:
$a_0=1,a_1=-1, a_n=\sum_{i=1}^{n-1} \dfrac{a_ia_{n-i}}{1-2^{1-n}},\, n\geq 2$
and
$ a_n=\left(\frac{-1}{\lambda^b}\right)^n\frac{\Gamma(nb+1)}{n!},$ for some suitable $b$ and $\lambda$. (Actually, we know what each of these has to be just from the initial values $a_1,a_2$.)
I've tried some futzing about with the series expansions of the gamma function, and treating these as associated to some generating functions, but I'm running into walls. Some hints would be greatly appreciated.