Construct a function $\displaystyle f(x)\in C^{\infty}(-\infty,+\infty)$

that satisfy $\displaystyle f^{(2n+1)}(0) = n$ and $\displaystyle f^{(2n)}(0) = 0$,for all $\displaystyle n\geq0$.

Furthermore,show that there are infinity $\displaystyle f(x)$satisfy the condition above