A note on a definite integral of combinatorial interest

In this note, we consider the numerical sequence {I^(s)_n}_{n≥0}, where I^(s)_n is the integral of the n-th power of the trinomial 1 + sx + x^2 on the interval [0, 1]. The integrals I^(s)_n can be easily expressed in terms of the generalized trinomial coefficients. We show that they can also be expressed in several other different, non-trivial, ways as a single binomial sum. Moreover, we show that the integrals I^(s)_n, when positive, form a logconvex sequence. Finally, we define a new class of polynomials I^(s)_n(x) as a natural extension of the integrals I^(s)_n and we show that, for every fixed s, they form an Appell sequence