Proofs of two conjectures on Catalan triangle numbers

Abstract

We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana et al. [Discrete Math. 340 (2017), 2388–2397]. The first one is proved by using Zeilberger's algorithm, and the second one is proved by establishing its q-analogue.

Keywords:

• Catalan numbers
• Catalan triangle
• Zeilberger's algorithm
• q-Chu-Vandermonde summation
• cyclotomic polynomial

MR Subject Classifications:

• 05A30
• 05A10
• 11B65

Acknowledgements

The authors thank the anonymous referee for a careful reading of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Victor J. W. Guo  http://orcid.org/0000-0002-4153-715X

Additional information

Funding

The first author was partially supported by the National Natural Science Foundation of China (grant 11771175), the Natural Science Foundation of Jiangsu Province (grant BK20161304), and the Qing Lan Project of Education Committee of Jiangsu Province.