Let $\mathbb{Z}_{\geq 0}[|t|]$ be the ring of power series with non-negative integer coefficients and consider the power series
$$P(t) = \sum_{i=0}^ \infty a_i t^i \in \mathbb{Z}_{\geq 0}[|t|]$$
$$P^2(t) = \left( \sum_{i=0}^ \infty a_i t^i\right)\left( \sum_{i=0}^ \infty a_i t^i\right) = \sum_{i=0}^ \infty b_i t^i \in \mathbb{Z}_{\geq 0}[|t|]$$ s.t. $$\limsup_{n \rightarrow \infty} \frac{b_n}{a_n} = + \infty.$$
Let $c_i \in \mathbb{Z}$ for $0\leq i \leq m$ s.t. $\sum_{j=0}^{m} c_j a_{n-j} \in \mathbb{N}$ for each $n \in \mathbb{N}.$
Can we conclude that
$$\limsup_{n \rightarrow \infty} \frac{\sum_{j=0}^{m} c_j b_{n-j}}{\sum_{j=0}^{m} c_j a_{n-j}} = + \infty\;?$$
Please let me know if there is any information needed. Any help is highly appreciated. Thanks in advance.
