104
Views
0
CrossRef citations to date
0
Altmetric
Articles

Proofs of two conjectures on Catalan triangle numbers

ORCID Icon &
Pages 1473-1487 | Received 08 Oct 2017, Accepted 05 Jun 2018, Published online: 07 Aug 2018

References

  • G.E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, 1998.
  • X. Chen and W. Chu, Moments on Catalan numbers, J. Math. Anal. Appl. 349 (2009), pp. 311–316. doi: 10.1016/j.jmaa.2008.08.042
  • W.Y.C. Chen and Q.-H. Hou, Factors of the Gaussian coefficients, Discrete Math. 306 (2006), pp. 1446–1449. doi: 10.1016/j.disc.2006.03.031
  • W. Chu, Moments on quadratic binomial products, J. Number Theory 178 (2017), pp. 19–30. doi: 10.1016/j.jnt.2017.02.005
  • G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
  • V.J.W. Guo and C. Krattenthaler, Some divisibility properties of binomial and q-binomial coefficients, J. Number Theory 135 (2013), pp. 167–184. doi: 10.1016/j.jnt.2013.08.012
  • V.J.W. Guo and S.-D. Wang, Factors of sums involving q-binomial coefficients and powers of q-integers, J. Diff. Equ. Appl. 23 (2017), pp. 1670–1679.
  • V.J.W. Guo and S.-D. Wang, Factors of sums and alternating sums of products of q-binomial coefficients and powers of q-integers, Taiwanese J. Math. doi:10.11650/tjm/180601.
  • V.J.W. Guo and J. Zeng, Some arithmetic properties of the q-Euler numbers and q-Salié numbers, Eur. J. Combin. 27 (2006), pp. 884–895. doi: 10.1016/j.ejc.2005.04.009
  • V.J.W. Guo and J. Zeng, Factors of binomial sums from the Catalan triangle, J. Number Theory 130 (2010), pp. 172–186. doi: 10.1016/j.jnt.2009.07.005
  • V.J.W. Guo and J. Zeng, Factors of sums and alternating sums involving binomial coefficients and powers of integers, Int. J. Number Theory 7 (2011), pp. 1959–1976. doi: 10.1142/S1793042111004812
  • J.M. Gutiérrez, M.A. Hernández, P.J. Miana and N. Romero, New identities in the Catalan triangle, J. Math. Anal. Appl. 341 (2008), pp. 52–61. doi: 10.1016/j.jmaa.2007.09.073
  • E. Kilic and H. Prodinger, Identities with squares of binomial coefficients: An elementary and explicit approach, Publ. Inst. Math. 99(113) (2016), pp. 243–248. doi: 10.2298/PIM1613243K
  • D. Knuth and H. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math. 396 (1989), pp. 212–219.
  • W. Koepf, Hypergeometric Summation—An Algorithmic Approach to Summation and Special Function Identities, 2nd Ed., Springer, London, 2014.
  • P.J. Miana, H. Ohtsuka and N. Romero, Sums of powers of Catalan triangle numbers, Discrete Math. 340 (2017), pp. 2388–2397. doi: 10.1016/j.disc.2017.05.006
  • P.J. Miana and N. Romero, Computer proofs of new identities in the Catalan triangle, in: Proc. of the “Segundas Jornadas de Teoría de Números”, Madrid, 2007, Biblioteca de la Revista Matemática Iberoamericana, pp. 203–208.
  • P.J. Miana and N. Romero, Moments of combinatorial and Catalan numbers, J. Number Theory 130 (2010), pp. 1876–1887. doi: 10.1016/j.jnt.2010.01.018
  • M. Petkovšek, H.S. Wilf and D. Zeilberger, A=B, A K Peters, Ltd., Wellesley, MA, 1996.
  • L.W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), pp. 83–90. doi: 10.1016/0012-365X(76)90009-1
  • S. Stanimirović, P. Stanimirović and A. Ilić, Ballot matrix as Catalan matrix power and related identities, Discrete Appl. Math. 160 (2012), pp. 344–351. doi: 10.1016/j.dam.2011.10.016
  • R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999.
  • R.P. Stanley, Catalan Numbers, Cambridge University Press, Cambridge, 2015.
  • Y. Sun and F. Ma, Some new binomial sums related to the Catalan triangle, Electron. J. Combin. 21(1) (2014), pp. #P1.33.
  • Z. Zhang and B. Pang, Several identities in the Catalan triangle, Indian J. Pure Appl. Math. 41 (2010), pp. 363–378. doi: 10.1007/s13226-010-0022-0

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.