Consider $\mathbb{C}^n$ with coordinates $x_1,\ldots,x_n$ and let $\partial_1,\ldots,\partial_n$ be the corresponding vector fields. Then the canonical free rank 1 D-module should be $$ \mathbb{C}[x_1,\ldots ,x_n][\partial_1,\ldots, \partial_n] $$
Why is it if we have a polynomial $\phi$, the following equation holds $$ \phi\cdot\partial^\alpha = \sum_{0\leq \beta \leq \alpha} \frac{\alpha!}{(\alpha - \beta)!\beta!}(-1)^{|\alpha - \beta|}\partial^\beta \cdot\partial^{\alpha - \beta}(\phi) $$
From my understanding, a D-module structure on a module $M$ is just a lie algebra homomorphism $$ D \to \text{End}_\mathbb{C}(M) $$
That's (Möbius) inversion applied to the product rule.