A continuation from my two previous posts:

I have got the following recurrence which describes polynomials:

$$ C_n(a) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} a^{t(n-t)} C_t(a) $$ where $C_1(a)=C_0(a) = 1$. The ultimate goal is to prove (this is the conjecture) that these polynomials tend to 1 pointwise for $0 \leq a < 1$.

What I got so far, is the generation function approach. By denoting $A_n(a) = a^{-\binom{n}{2}}C_n(a)$ I got the recurrence $$ A_n(a) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} a^{-\binom{n-t}{2}}A_t(a) $$, and so the generating function for $A_n(a)$ is $$ F(x) = e^{g(x)} $$ where $g$ is itself a generating function $$ g(x) = \sum\limits_{t=1}^{\infty} a^{-\binom{t}{2}}\frac{x^t}{t!} $$

My questions are

1) what's next in order to discover asymptotics of $C_n$? Evaluate poles of $F(z)$?

2) is the generating function $g(x)$ studied? I know that it is when the coefficient is $q^{\binom{n}{2}}$ for $ 0 < q < 1$, but in my case it is greater than 1.

Also, am I missing something which my help a lot?

Thank you!

UPDATE 1:

I was able to reformulate the problem as ($b > 1$) $$ X_n(b) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} b^{\binom{n-t}{2}} X_t(b) $$ Prove that $|X_n(b) - b^{\binom{n}{2}}| \to 0$. So, here the generating function might work?