{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T04:35:47Z","timestamp":1766378147789,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T00:00:00Z","timestamp":1669766400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Romanian Ministry of Education","award":["UTCB-CDI-2022-008"],"award-info":[{"award-number":["UTCB-CDI-2022-008"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to every pair of nonnegative integers (except for the origin). We study the geometric properties of this surface and prove that it contains exactly five half-lines. The mean curvature and the Gauss curvature of the surface are also calculated.<\/jats:p>","DOI":"10.3390\/axioms11120685","type":"journal-article","created":{"date-parts":[[2022,11,30]],"date-time":"2022-11-30T05:45:22Z","timestamp":1669787122000},"page":"685","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On a Surface Associated to the Catalan Triangle"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2718-8133","authenticated-orcid":false,"given":"Marilena","family":"Jianu","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4612-9286","authenticated-orcid":false,"given":"Sever","family":"Achimescu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2770-023X","authenticated-orcid":false,"given":"Leonard","family":"D\u0103u\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5002-7963","authenticated-orcid":false,"given":"Ion","family":"Mierlu\u015f-Mazilu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2033-8394","authenticated-orcid":false,"given":"Adela","family":"Mihai","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"},{"name":"Interdisciplinary Doctoral School, Transilvania University of Bra\u015fov, 500036 Bra\u015fov, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7736-9648","authenticated-orcid":false,"given":"Daniel","family":"Tudor","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Stanley, R. 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An Introduction to Probability Theory and Its Applications, John Wiley & Sons. [3rd ed.]."},{"key":"ref_7","unstructured":"Sloane, N.J.A. (2022, November 29). The On-Line Encyclopedia of Integer Sequences (OEIS). Available online: https:\/\/oeis.org."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0012-365X(76)90009-1","article-title":"A Catalan triangle","volume":"14","author":"Shapiro","year":"1976","journal-title":"Discrete Math."},{"key":"ref_9","unstructured":"D\u0103u\u015f, L., Jianu, M., Beiu, R.M., and Beiu, V. (2020, January 27\u201329). A tale of Catalan triangles\u2014Counting lattice paths. Proceedings of the 9th International Workshop on Soft Computing Applications\u2014SOFA 2020, Arad, Romania."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1876","DOI":"10.1016\/j.jnt.2010.01.018","article-title":"Moments of combinatorial and Catalan numbers","volume":"130","author":"Miana","year":"2010","journal-title":"J. Number Theory"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Beiu, V., D\u0103u\u015f, L., Jianu, M., Mihai, A., and Mihai, I. (2022). On a surface associated with Pascal\u2019s triangle. Symmetry, 14.","DOI":"10.3390\/sym14020411"},{"key":"ref_12","first-page":"4865","article-title":"Approximating the level curves on Pascal\u2019s surface","volume":"17","author":"Jianu","year":"2022","journal-title":"Int. J. Comput. Commun. Control"},{"key":"ref_13","unstructured":"Abramowitz, M., and Stegun, I.A. (1972). Gamma function and related functions. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover. [10th ed.]."},{"key":"ref_14","unstructured":"do Carmo, M. (1976). Differential Geometry of Curves and Surfaces, Prentice-Hall."},{"key":"ref_15","unstructured":"Struik, D.J. (1961). Lectures on Classical Differential Geometry, Dover Publications. [2nd ed.]."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/685\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:29:57Z","timestamp":1760146197000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/685"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,30]]},"references-count":15,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120685"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120685","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,11,30]]}}}