The Sagan-Savage Lucas-Catalan Polynomials Have Positive Coefficients

Shalosh B. EKHAD^{1}

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 conjectured that they are in fact polynomials with 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].

References

[1] Bruce Sagan, 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 .

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

1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen

Rd., Piscataway, NJ 08854-8019, USA. c/o zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg/ekhad.html . Jan. 17, 2011. Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger .