A Note on the Conjecture of Ramirez and Sirvent

Emrah Kılıç Mathematics Department TOBB University of Economics and Technology 06560 Ankara Turkey

                       ekilic@etu.edu.tr

Helmut Prodinger Department of Mathematical Sciences Stellenbosch University 7602 Stellenbosch South Africa

                      hproding@sun.ac.za

Abstract We give a proof of a recent conjecture of Ramirez and Sirvent on the generating

function of the incomplete Tribonacci numbers.

1 Introduction

In a recent paper of Ramirez and Sirvent [1], the authors have defined the incomplete Tribonacci sequence of numbers and polynomials. They have also studied recurrence relations, some properties of these numbers and polynomials, and the generating function of the incomplete Tribonacci numbers.

Let

a_{n} = \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ i=0 \end{matrix} ∑_{j=0}^{i} i\binom{i}{j}\binom{n − i − j − 1}{i}.

Ramirez and Sirvent conjectured that the generating function of the sequence {a_{n}}_{n=0}^{∞} is

∑_{n≥0} a_{n}z^{n} = \frac{z^{3} + z^{4}}{(1 − z − z^{2} − z^{3})^{2}} .

In this short note, we give a proof of this conjecture.

2 A proof of the conjecture

Let A (z) be the generating function of the sequence {a_{n}}_{n=0}^{∞} . Now consider

A (z) = ∑_{0≤j≤i} i\binom{i}{j}z^{i+j+1} ∑_{n≥i} \binom{n − i − j − 1}{i}z^{n−i−j−1}

= ∑_{0≤j≤i} i\binom{i}{j}z^{i+j+1} ∑_{n≥i} \binom{n}{i}z^{n}

= ∑_{0≤j≤i} i\binom{i}{j}z^{i+j+1} \frac{z^{i}}{(1 − z)^{i+1}} = ∑_{0≤j≤i} i\binom{i}{j}\frac{z^{2i+j+1}}{(1 − z)^{i+1}}

= ∑_{i≥0} i\frac{z^{2i+1}}{(1 − z)^{i+1}} ∑_{0≤j≤i} \binom{i}{j}z^{j}

= ∑_{i≥0} i\frac{z^{2i+1}}{(1 − z)^{i+1}} (1 + z)^{i}

= \frac{z}{1 − z} ∑_{i≥0} i (\frac{z^{2} (1 + z)}{1 − z})^{i}

= \frac{z}{1 − z} \frac{z^{2}(1 − z^{2})}{(1 − z − z^{2} − z^{3})^{2}}

= \frac{z^{3} + z^{4}}{(1 − z − z^{2} − z^{3})^{2}} ,

which proves the conjecture.

References

[1] J. L. Ramírez and V. F. Sirvent, Incomplete Tribonacci numbers and polynomials, J. Integer Sequences, 17 (2014), Article 14.4.2.

2010 Mathematics Subject Classification: Primary 11B39. Keywords: Tribonacci number, generating function.

Received February 17 2014; revised version received March 27 2014; April 7 2014. Published in Journal of Integer Sequences, April 9 2014.

Return to Journal of Integer Sequences home page.