Suppose $f$ is an unknown polynomial of degree $n$ (in one indeterminate) but the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ is given. It is a nice exercise to show that one needs only the first $n+1$ terms of the sequence to determine the coefficients of $f$. That is, simply solve the matrix equation $A\mathbf{x} = b$, where $\mathbf{b} = (f(0), \dots, f(n))^{\top}$, $A$ is the Vandermonde matrix of $(i^{j})_{i,j = 0, \dots, n}$ and $\mathbf{x} = (c_{0}, \dots, c_{n})^{\top}$ (the unknown coefficients of $f$).
Question: Is there a closed form expression for the coefficients of a finite degree polynomial $f$ in terms of the sequence $\{ f(k) \}_{k \in \mathbb{N}}$ that doesn't involve matrix inversion or differentiation or explicitly calculating the polynomial in question?
(Motivation) The Ehrhart polynomial counts the number of integer lattice points intersecting a dilate of a polytope and can be calculated by the residue of an associated complex rational function (see M. Beck's articles on the subject). Some of the coefficients of the Ehrhart polynomial can be related to an $n$-volume, a relative area and the euler characteristic of said polytope. However, computing coefficients of the Ehrhart polynomial is not a particularly easy task. Having simple formulas for them, say in terms of the residues above would be nice to have at one's disposal. A reasonable starting point is answering the question above.
Thanks!