calculus - Find the function that has the following series representation

I encounter this series in an old book of infinite series. The series is as followed:

$1+(\dfrac{1}{2})x^2+(\dfrac{1\cdot 3}{2\cdot4})^2x^4+(\dfrac{1\cdot3\cdot5}{2\cdot4\cdot6})^3x^6+(\dfrac{1\cdot3\cdot5\cdot7}{2\cdot4\cdot6\cdot8})^4x^8...$

Judging from the series, it must be an even function.
Is it possible to find the closed form function for this series. Also, instead of the power of 2 in coefficients, we may have them in n. Do we have a general formula to generate the functions for power of n?

I try to differentiate this series but it doesn't seem to produce anything fruitful.