Tuenter二项式和的推广
Generalising Tuenter's binomial sums
摘要：
Tuenter [Fibonacci Quarterly 40（2002），175-180]和其他作者考虑了\[S_r(n) = \sum_k \binom{2n}{k}|n-k|^r,\]形式的中心二项式和，其中$r$和$n$是非负整数。我们认为总和的形式\[U_r(n) = \sum_k \binom{n}{k}|n/2-k|^r\]这是一个概括的Tuenter的总和为$S_r(n) = U_r(2n)$，但$U_r(n)$也是定义良好的奇数参数$n$。$U_r(n)$可以被解释为具有$n$步长的对称伯努利随机游动的矩。$U_r(n)$的形式取决于$r$和$n$的奇偶性。事实上，$U_r(n)$是多项式（取决于$r$和$n$的奇偶校验）乘以2的幂或二项式系数的乘积。在所有情况下，多项式都可以用Dumont-Foata多项式表示。我们给出了函数$U_r(n)$和/或相关多项式的递推关系、生成函数和显式公式。
英文摘要：
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.
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