Asymptotic behavior (with respect to the number of trials) of symmetric generalizations of binomial distributions and their related entropies are studied through three examples. The first one derives from the q-exponential as a generating function. The second one involves the modified Abel polynomials, and the third one involves Hermite polynomials. The former and the latter have extensive Boltzmann-Gibbs whereas the second one (Abel) has extensive Renyi entropy. A probabilistic model is presented for this exceptional case.