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\vskip 1cm{\LARGE\bf 
A Note on the Conjecture of Ramirez and \\
\vskip .1in
Sirvent
}
\vskip 1cm
\large
Emrah K\i l\i \c{c}\\
Mathematics Department\\
TOBB University of Economics and Technology \\
06560 Ankara\\
Turkey\\
\href{mailto:ekilic@etu.edu.tr}{\tt ekilic@etu.edu.tr} \\
\ \\
Helmut Prodinger\\
Department of Mathematical Sciences\\
Stellenbosch University\\
7602 Stellenbosch\\
South Africa\\
\href{mailto:hproding@sun.ac.za}{\tt hproding@sun.ac.za} \\
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\begin{abstract}
We give a proof of a recent conjecture of Ramirez and Sirvent on the
generating function of the incomplete Tribonacci numbers.
\end{abstract}

\section{Introduction}

In a recent paper of Ramirez and Sirvent \cite{RS}, 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
\begin{equation*}
a_{n}=\sum_{i=0}^{\left \lfloor \frac{n-1}{2}\right \rfloor }\sum_{j=0}^{i}i%
\dbinom{i}{j}\dbinom{n-i-j-1}{i}.
\end{equation*}
Ramirez and Sirvent conjectured that the generating function of the sequence $\left \{
 a_{n}\right \} _{n=0}^{\infty }$ is
\begin{equation*}
\sum_{n\ge0}a_nz^n=\frac{z^{3}+z^{4}}{\left( 1-z-z^{2}-z^{3}\right) ^{2}}.
\end{equation*}
In this short note, we give a proof of this conjecture.

\section{A proof of the conjecture}

 Let $A\left(
z\right) $ be the generating function of the sequence $\left \{ a_{n}\right \}
_{n=0}^{\infty }.$
%
Now consider
\begin{align*}
A\left( z\right) &amp; =\sum_{0\leq j\leq i}i\dbinom{i}{j}z^{i+j+1}\sum_{n\geq i}%
\dbinom{n-i-j-1}{i}z^{n-i-j-1} \\
&amp; =\sum_{0\leq j\leq i}i\dbinom{i}{j}z^{i+j+1}\sum_{n\geq i}\dbinom{n}{i}%
z^{n} \\
&amp; =\sum_{0\leq j\leq i}i\dbinom{i}{j}z^{i+j+1}\frac{z^{i}}{\left( 1-z\right)
^{i+1}}=\sum_{0\leq j\leq i}i\dbinom{i}{j}\frac{z^{2i+j+1}}{\left(
1-z\right) ^{i+1}} \\
&amp; =\sum_{i\geq 0}i\frac{z^{2i+1}}{\left( 1-z\right) ^{i+1}}\sum_{0\leq j\leq
i}\dbinom{i}{j}z^{j} \\
&amp; =\sum_{i\geq 0}i\frac{z^{2i+1}}{\left( 1-z\right) ^{i+1}}\left( 1+z\right)
^{i} \\
&amp; =\frac{z}{1-z}\sum_{i\geq 0}i\left( \frac{z^{2}\left( 1+z\right) }{1-z}%
\right) ^{i} \\
&amp; =\frac{z}{1-z}\frac{z^{2}(1-z^{2})}{\left( 1-z-z^{2}-z^{3}\right) ^{2}} \\
&amp; =\frac{z^{3}+z^{4}}{\left( 1-z-z^{2}-z^{3}\right) ^{2}},
\end{align*}%
which proves the conjecture.

\bigskip

\begin{thebibliography}{9}
\bibitem{RS} J. L. Ram\'{\i}rez and V. F. Sirvent,
Incomplete {T}ribonacci numbers
and polynomials,  {\it J. Integer Sequences}, {\bf 17} (2014), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL17/Ramirez/ramirez4.html}{Article 14.4.2}.
\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B39.

\noindent \emph{Keywords: } Tribonacci number, generating function.

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\vspace*{+.1in}
\noindent
Received February 17 2014;
revised version received  March 27 2014; April 7 2014.
Published in {\it Journal of Integer Sequences}, April 9 2014.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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