It is well known that the the special mathematical constant e can be written in the form e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\cdots. With this in mind, determine the value of
\sum_{j=3}^{\infty} \frac{j}{\left\lfloor\frac{j}{2}\right\rfloor!}.
Express your answer in terms of e.