A remarkable sequence of integers

Abstract

A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.The usual elementary proof of (2.5) presented in textbooks is to produce a recurrence for J 2,m . Writing cos 2 θ = 1 − sin 2 θ and using integration by parts yieldsNow verify that the right side of (2.5) satisfies the same recursion and that both sides give π/2 for m = 0.

Partially 2-colored permutations and the Boros–Moll polynomials

“…Chen and Xia [10] have proved a stronger property of d i (m), called the ratio monotone property, which implies both the log-concavity and the spiral property. Moll [14,15] posed a conjecture that is stronger than the log-concavity of P m (a). This conjecture has been proved by Chen and Xia [11].…”

Series evaluation of a quartic integral

“…In a beautiful personal story [6] Victor Moll describes his encounter with certain quartic integral and derives its evaluation and goes on to study analytic and number theoretic properties (log-concavity, p-adic valuations, location of the zeros, etc.) of a polynomial associated with the evaluation of the integral [1,2,4,5,6,7]. In this article we use the Almkvist-Zeilberger algorithm ( [3,8,9,10]) to derive a new series evaluation of this integral.…”