Generalising Tuenter's binomial sums

Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form $$S_r(n)=\sum _k\left(\genfrac{}{}{0px}{}{2n}{k}\right)\mathrm{\mid }n-k{\mathrm{\mid }}^r,$$ where $r$ and $n$ are non-negative integers. We consider sums of the form $$U_r(n)=\sum _k\left(\genfrac{}{}{0px}{}{n}{k}\right)\mathrm{\mid }n\mathrm{/}2-k{\mathrm{\mid }}^r$$ which are a generalisation of Tuenter’s sums as $S_r(n)=U_r(2n)$ but $U_r(n)$ is also well-defined for odd arguments $n$. $U_r(n)$ may be interpreted as a moment 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.