inequality for series

Question

Let $j \in Z_+$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.

Let $X(i)=|a^{(2i)}_j|j!$. Verify that $X(i)\leq X(1)$ for $j\geq 4i$.

Thank you.