Polynomials having the generating function $\exp(x(e^{t_1}+e^{t_2}-2))$

Question

The generating function of the Touchard polynomials $T_n(x)$ is given by $$\sum_{n=0}^\infty \frac{T_n(x)}{n!}t^n = \exp(x(e^t-1)).$$ Is there a known set of polynomials having a generating function $$\exp(x(e^{t_1}+e^{t_2}-2))?$$ More general, $$\exp(x(\sum_{n=1}^d (e^{t_n}-1)))?$$