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