The expansion coefficients of classical Eulerian polynomials $A_n(t)$ are nonnegative by the symmetric basis of $\{t^i(1+t)^{n-2i}\}$, which is called the gamma-nonnegativity proved by Foata-Schzenberger in 1970. In this paper, we investigate the interlacing nonnegativity of the inverse gamma-expansion, that is, the coefficients of $\{(1+t)^n\}$ expanded by the basis of $\{(-t)^i A_{n-2i}(t)\}$ are nonnegative. In addition, we also prove that the coefficients of $\{(1+t)^n\}$ in the inverse gamma-expansion of several famous combinatorial polynomials such as Narayana polynomials are interlacing nonnegativitive. It is an interesting open problem to look for combinatorial interpretations of these coefficients.