  We present combinatorial and analytical results concerning a Sheffer sequence
with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$
is a quadratic polynomial with real zeros. By using the properties of Riordan
matrices we address combinatorial properties and interpretations of our Sheffer
sequence of polynomials and their coefficients. We also show that apart from
two exceptional zeros, the zeros of polynomials with large enough degree in
such a Sheffer sequence lie on the line $x=1/2+it$.
