Even generating function with Stirling numbers of the second kind

Question

Is there a known closed form for the following generating function? $$ \sum_{n=0}^{\infty}\sum_{k=0}^{n} \frac{S(n, k)}{(2n) !}x^{2n}y^k $$ If we had $n!$ at the denominator instead, this would simply be the well-known mixed generating function $e^{y(e^{x^2}-1)}$, but the $(2n)!$ changes everything. Any help would be greatly appreciated.