References

  1. Belbachir H., Belkhir A., Djellas I.-E. Permanent of Toeplitz-Hessenberg matrices with generalized Fibonacci and Lucas entries. Appl. Appl. Math. 2022, 17 (2), 558–570.
  2. Callan D. Some bijections and identities for the Catalan and Fine numbers. Sém. Lothar. Combin. 2005, 53, article B53e.
  3. Chen Z., Pan H. Identities involving weighted Catalan, Schröder and Motzkin paths. Adv. Appl. Math. 2017, 86, 81–98. doi:10.1016/J.AAM.2016.11.011
  4. Deng E.Y.P., Yan W.-J. Some identities on the Catalan, Motzkin and Schröder numbers. Discrete Appl. Math. 2008, 156 (14), 2781–2789. doi:10.1016/j.dam.2007.11.014
  5. Deutsch E. Dyck path enumeration. Discrete Math. 1999, 204, 167–202. doi:10.1016/S0012-365X(98)00371-9
  6. Deutsch E., Shapiro L. A survey of the Fine numbers. Discrete Math. 2001, 241 (1–3), 241–265. doi:10.1016/S0012-365X(01)00121-2
  7. Elouafi M. A unified approach for the Hankel determinants of classical combinatorial numbers. J. Math. Anal. Appl. 2015, 431 (2), 1253–1274. doi:10.1016/j.jmaa.2015.06.034
  8. Goy T., Shattuck M. Determinant formulas of some Toeplitz-Hessenberg matrices with Catalan entries. Proc. Indian Acad. Sci. Math. Sci. 2019, 129 (4), article number 46. doi:10.1007/s12044-019-0513-9
  9. Goy T., Shattuck M. Determinants of Toeplitz-Hessenberg matrices with generalized Fibonacci entries. Notes Number Theory Discrete Math. 2019, 25 (4), 83–95. doi:10.7546/nntdm.2019.25.4.83-95
  10. Goy T., Shattuck M. Determinant identities for Toeplitz-Hessenberg matrices with tribonacci number entries. Trans. Comb. 2020, 9 (2), 89–109. doi:10.22108/TOC.2020.116257.1631
  11. Goy T., Shattuck M. Some Toeplitz-Hessenberg determinant identities for the tetranacci numbers. J. Integer Seq. 2020, 23, article 20.6.8.
  12. Goy T., Shattuck M. Determinants of some Hessenberg-Toeplitz matrices with Motzkin number entries. J. Integer Seq. 2023, 26, article 23.3.4.
  13. Komatsu T., Ramı́rez J.L. Some determinants involving incomplete Fubini numbers. An. Şt. Univ. Ovidius Constanţa. Ser. Mat. 2018, 26 (3), 143–170. doi:10.2478/auom-2018-0038
  14. Merca M. A note on the determinant of a Toeplitz-Hessenberg matrix. Spec. Matrices 2013, 1, 10–16. doi:10.2478/spma-2013-0003
  15. Mu L., Wang Y. Hankel determinants of shifted Catalan-like numbers. Discrete Math. 2017, 340 (6), 1389–1396. doi:10.1016/j.disc.2016.09.035
  16. Muir T. The Theory of Determinants in the Historical Order of Development. Vol. 3. Dover Publications, 1960.
  17. Shapiro L.V., Wang C.J. A bijection between \(3\)-Motzkin paths and Schröder paths with no peak at odd height. J. Integer Seq. 2009, 12, article 09.3.2.
  18. Sloane N.J.A. et al. The On-Line Encyclopedia of Integer Sequences, 2023. Available at https://oeis.org
  19. Qi F. On negativity of Toeplitz-Hessenberg determinants whose elements contain large Schröder numbers. Palestine J. Math. 2022, 11 (4), 373–378.
  20. Qi F., Guo B.-N. Explicit and recursive formulas, integral representations, and properties of the large Schröder numbers. Kragujevac J. Math. 2017, 41 (4), 121–141.
  21. Qi F., Shi X.-T., Guo B.-N. Two explicit formulas of the Schröder numbers. Integers 2016, 16, article 23.