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.