Let $X \in [-n,n] \cap \mathbb Z$ be a sum of $n$ iid Rademacher random variables and let $X_1,\ldots,X_k$ be $k \ge 1$ iid copies of $X$, and define the product $Z_k := X_1 X_2 \ldots X_k$. I'm interested in the moment-generating function (MGF) of $Z$, i.e $\eta_k(t) := \mathbb E[e^{-tZ_k}]$. Note that we can write $X = 2B - n$, where $B$ is a Binomial r.v with parameter $n$ and $1/2$. Thus, by marginalizing w.r.t $X_1$, we have
$$ \begin{split} \eta_k(t) &= \mathbb E_{X_2,\ldots,X_k} \mathbb E_B[e^{-t(2B-n) Z_{k-1}}]= 2^{-n}\sum_{j=0}^n {n \choose j}\mathbb E_{X_2,\ldots,X_k} [e^{-t(2j-n)Z_{k-1}}]\\ &= 2^{-n}\sum_{j=0}^n {n \choose j}\eta_{k-1}((2j-n)t). \end{split} $$
This gives a recursion formula for $\eta_k(t)$. The base case is $\eta_1(t) = \mathbb E [e^{-tX_1}] = \cosh^n(t)$.
Question 1. What is a simple analytic formula for $\eta_k(t)$ (i.e solve the above recursion) ?
N.B.: If I knew the generating $z \mapsto G(z,t):=\sum_{k \ge 0} \eta_k(t) z^k$ function of the sequence $(\eta_k(t))_{k \ge 0}$, I could extract it like so: $\eta_k(t) = [z^k] G(z,t)$.
Question 2. As a function of $n$ and $t \gt 0$, how fast does $\eta_k(t)$ grow ? If it helps, you may assume $n \ge C \log k$ for some sufficiently large constant $C$. You may even assume $\log n \asymp \log k$. Is $\eta_k(t)$ bounded in any of these regimes ?