Analytic Aspects of the Generalized Second-Order Central Trinomial Coefficients

The generalized second-order central trinomial coefficients Tn(2)(b,c) are a special case of the generalized r-order central trinomial coefficients, corresponding to r=2. We show that all zeros of the polynomial T(2)n(x)=∑k=0⌊n/2⌋T(2)(n,k)xk are real. And zeros of T(2)n−1(x) interlace those of T(2)n−2(x), as well as those of T(2)n(x). Using this result, we also discuss the asymptotic normality of T(2)(n,k).