Derivatives of delta function as a basis for distributions

Question

Is there some sense in which one could write any distribution as a sum of this sort?

$$A(x,y)=\sum_{n=0}^{\infty}a_n(x)i^n\frac{\partial^n}{\partial x^n}\delta (x-y)$$

Provided that the rhs acting on a test function is convergent for all $x$.