A Family of Palindromic Polynomials

The relation between the roots of a polynomial P (z) and its coefficients was one of the driving forces in the development of Algebra. The desire to obtain a closed-form expression for the roots of P (z) = 0 ended with a negative solution at the hands of N. Abel and E. Galois at the beginning of the 19 century: a generic polynomial equation of degree 5 or more cannot be solved by radicals. The reader should be aware of the existence of analytic expressions for the roots of a polynomial. Naturally these involve non-algebraic functions: for the quintic equation this is done using elliptic functions, as explained in the beautiful text [3], and for higher degree the formulas involve the so-called theta functions. In spite of this set-back, the study of roots of polynomials has continued throughout the centuries. Classical results include Newton’s statement that if P has only negative real roots then the coefficients aj of P form a logconcave sequence, that is, aj − aj−1aj+1 ≥ 0. For starters, the reader is refered to [4]. Logconcave sequences arise in many combinatorial contexts, the simplest of which are the binomial coefficients {( n k ) : 0 ≤ k ≤ n }