Let $\{a_n\}_{n\in\mathbb{N}}$ be an increasing sequence of natural numbers, and $ f_A(x)=\sum_{n\in\mathbb{N}}\frac{x^{a_n}}{a_n!}. $ There are some cases in which the limit $ l_A=\lim_{x\to+\infty} \frac{1}{x}\,\log(f_A(x)) $ does not exist. However, if $\{a_n\}_{n\in\mathbb{N}}$ is an arithmetic progression, we have $l_A=1$ (it follows from a straightforward application of the discrete Fourier transform). Consider now the case $a_n=n^2.$
Is it true that there exists a positive constant $c$ for which $\forall x>0,\quad e^{-x}f_A(x)=\sum_{k\in\mathbb{N}}x^k\left(\sum_{0\leq j\leq\sqrt{k}}\frac{(-1)^{k-j^2}}{(j^2)!\,(k-j^2)!}\right)\geq c\;?$
Is it true that $l_A=1$?