Here is an explicit formula for your ratio $r_n=\frac{n_n}{d_n}$:
$$r_n=
\frac{\sum_{k=0}^n(-1)^k\binom{n+k}{2k}x^k}
{\sum_{k=0}^n(-1)^k\binom{n+k+1}{2k+1}x^k}.$$