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\bf
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{
The Sagan-Savage Lucas-Catalan Polynomials Have Positive Coefficients
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\centerline{ {\it
Shalosh B.
EKHAD}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
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{\eighttt c/o zeilberg  at math dot rutgers dot edu} ,
\hfill \break
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg/ekhad.html} .
Jan. 17, 2011.
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger .
}
}

In the last few minutes of Bruce Sagan's wonderful talk[1]
about his joint work with Carla Savage[2] (whose notation I shall use),
he mentioned that they can prove that 
$\frac{1}{\{ n+1 \} } \left \{ { {2n} \atop {n} } \right \}$
are polynomials with integer coefficients, and {\it conjectured} that they are in fact
polynomials with {\it positive} coefficients.

But this follows immediately from the identity
$
\frac{1}{\{ n+1 \} } \left \{ { {2n} \atop {n} } \right \}=\left \{ {{2n-1} \atop {n-1} } \right \}+t\left \{ { {2n-1} \atop {n-2} } \right \}$
that after routine cancellations is equivalent to
$\{ 2n \}= \{n+1 \} \{ n \} + t \{n-1 \} \{ n \}$, that is the case $m=n$ of Lemma 2.1 of [2]. \halmos


{\bf References}

[1] Bruce Sagan, {\it Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences},
talk at the Rutgers University Experimental Mathematics seminar on Dec. 9, 2010,
videotaped by Edinah Gnang.  http://www.youtube.com/watch?v=Fdn890jg2U0 \quad .

[2]  Bruce Sagan and Carla Savage, {\it Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences},
Integers {\bf 10} (2010), 697-703, A52.  http://arxiv.org/abs/0911.3159 \quad .

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