Generalizing tuenter’s binomial sums

Uenter considered centered binomial sums of the form ()  where r and n are non-negative integers. We consider sums of the form which are a generalization of Tuenter’s sums and may be interpreted as moments of a symmetric Bernoulli random walk with n steps. The form of U_r (n) depends on the parities of both r and n. In fact, U_r (n) is the product of a polynomial (depending on the parities of r and n) times a power of two or a binomial coefficient. In all cases the polynomials can be expressed in terms of Dumont-Foata polynomials. We give recurrence relations, generating functions and explicit formulas for the functions U_r (n) and/or the associated polynomials.