Calculate the series $\sum_{n=1}^{\infty} \frac{x^n}{(3n)!}$

Question

I can't find the series $$\sum_{n=1}^{\infty} \frac{x^n}{(3n)!}$$

But I have no idea how to find.

Thanks for any hints or solutions.

Answer 1

Hint. A standard trick is to use $$ \alpha=-\frac{1}{2}+\frac{\sqrt{3}}{2}i ,\quad \alpha^3=1,\quad 1+\alpha+\alpha^2=0, $$ then consider $$ e^{x}+e^{\alpha x}+e^{\alpha^2 x}=\sum_{n=0}^{\infty}(1+\alpha^n+\alpha^{2n}) \frac{x^n}{n!}. $$