Represent expression in another basis

Question

Suppose there is a known analytic function $f(x)$.

Can we represent $f(t)$ in the form

$$f(0)+\sum_{n=2}^\infty a_n\frac{(t+1/2)^{n}-(t-1/2)^{n}}{n!}$$

?

In other words, how to find the coefficients $a_n$?