We propose a new randomized algorithm for multiplication in the ring of non-commutative polynomials $K[x_1,\dots,x_n]<\delta_1,\dots,\delta_n>$, where $\delta_i = x_i(\frac{\partial}{\partial x_i})$, dedicated to sparse inputs. The complexity of our algorithm is polynomial in the input size and on an a priori sparsity bound for the output.