Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.2

Generalizing Tuenter’s Binomial Sums

Richard P. Brent
Mathematical Sciences Institute
Australian National University
Canberra, ACT 2614


Tuenter considered centered binomial sums of the form

\begin{displaymath}S_r(n) = \sum_{k=0}^{2n} \binom{2n}{k}\vert n-k\vert^r,\end{displaymath}

where r and n are non-negative integers. We consider sums of the form

\begin{displaymath}U_r(n) = \sum_{k=0}^n \binom{n}{k}\vert n/2-k\vert^r,\end{displaymath}

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 Ur(n) depends on the parities of both r and n. In fact, Ur(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 Ur(n) and/or the associated polynomials.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000142 A000364 A001147 A001469 A001813 A002105 A009843 A036970 A047053 A054879 A083061 A160485 A245244.)

Received July 16 2014; revised versions received January 18 2015; January 25 2015. Published in Journal of Integer Sequences, January 26 2015.

Return to Journal of Integer Sequences home page