Putnam 2024 A6

Let $c_0,c_1,c_2,\dots$ be the sequence defined so that $$\frac{1-3x-\sqrt{1-14x+9x^2}}{4} = \sum_{k=0}^\infty c_k x^k$$ for sufficiently small $x$. For a positive integer $n$, let $A$ be the $n$-by-$n$ matrix with $i,j$-entry $c_{i+j-1}$ for $i$ and $j$ in $\{1,\dots,n\}$. Find the determinant of $A$.