Generalising Tuenter's binomial sums

TL;DR

This paper generalizes Tuenter's binomial sums to include odd arguments, explores their properties, and expresses related polynomials using Dumont-Foata polynomials, providing recurrence relations, generating functions, and explicit formulas.

Contribution

It introduces a new class of sums $U_r(n)$ extending Tuenter's sums to odd $n$, and characterizes their polynomial structure and generating functions.

Findings

$U_r(n)$ depends on the parity of $r$ and $n$

Polynomials are expressed via Dumont-Foata polynomials

Recurrence relations and explicit formulas are derived

Abstract

Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form \[S_r(n) = \sum_k \binom{2n}{k}|n-k|^r,\] where $r$ and $n$ are non-negative integers. We consider sums of the form \[U_r(n) = \sum_k \binom{n}{k}|n/2-k|^r\] 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.