Consider the two sequences $$a(n)=\sum_{k=1}^n\binom{n}k\sum_{j=1}^{k/2}\binom{k}{2j}\frac{(2j)!}{j!}$$ and $$b(n)=\sum_{k=0}^n\binom{n}k^2k!$$

**QUESTION.** Is this true?
$$\frac{a(n)}{b(n)}\longrightarrow 0 \qquad \text{as} \qquad n\rightarrow\infty.$$