A curious identity in connection with saddle-point method and Stirling's formula

We prove the curious identity in the sense of formal power series: $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[y^m]\exp (-\frac{t^2}{2}+\sum _{j\ge 3}\frac{(it{)}^j}{j!}{y}^{j-2})\mathrm{d}t={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[y^m]\exp (-\frac{t^2}{2}+\sum _{j\ge 3}\frac{(it{)}^j}{j}{y}^{j-2})\mathrm{d}t,$$ for $m=0,1,\dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor expansion of $f$. The generality of this identity from the perspective of saddle-point method is also examined.