Many polynomials can be generated by successive differentiations of a given base function. In particular, the second-order Eulerian polynomials $C_n(x)$ can be defined by \begin{align*} \left(\frac{x}{1-x}\frac{d}{dx}\right)^n\frac{x}{1-x}=\frac{C_n(x)}{(1-x)^{2n+1}}, \end{align*} which is well known to be the descent polynomial of Stirling permutations $\mathcal{Q}_n$. In this paper, we study an alternative differentiation and set \begin{align*} \left(\frac{d}{dx}\frac{x}{1-x}\right)^n\frac{x}{1-x}=\frac{\widetilde{C}_n(x)}{(1-x)^{2n+1}}. \end{align*} Then we provide combinatorial interpretations for the coefficients of $\widetilde{C}_n(x)$ by defining the new statistics $\mathrm{tdes}$, $\mathrm{tasc}$, and $\mathrm{tplat}$ on modified Stirling permutations $\mathcal{Q}_{n+1}^{(1)}$ together with the statistic $\mathrm{bra}$ on signed permutations $\mathfrak{S}_n^B$. We further consider the ternary polynomial $\widetilde{C}_n(x,y,z)$, whose specialization at $y=z=1$ recovers $\widetilde{C}_n(x)$. Finally, we derive a context-free grammar for $\widetilde{C}_n(x,y,z)$ and propose some related conjectures.