Journal of Integer Sequences, Vol. 19 (2016), Article 16.3.7

Some Binomial Sums Involving Absolute Values

Richard P. Brent
Australian National University
Canberra, ACT 2600

Hideyuki Ohtsuka
Bunkyo University High School
1191-7, Kami, Ageo-City
Saitama Pref., 362-0001

Judy-anne H. Osborn
The University of Newcastle
Callaghan, NSW 2308

Helmut Prodinger
Stellenbosch University
7602 Stellenbosch
South Africa


We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form

\begin{displaymath}S_{\alpha,\beta}(n) := \sum_{k,\;\ell}\binom{2n}{n+k}\binom{2n}{n+\ell}
\vert k^\alpha-\ell^\alpha\vert^\beta,\end{displaymath}

obtaining new results in the cases $\alpha = 1, 2$. We show that there is a close connection between these double sums in the case $\alpha=1$ and the single centered binomial sums considered by Tuenter.

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(Concerned with sequences A166337 A254408 A268147 A268148 A268149 A268150 A268151 A268152.)

Received January 30 2016; revised version received March 3 2016. Published in Journal of Integer Sequences, April 6 2016.

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