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Supercongruences involving Motzkin numbers and central trinomial coefficients

Part of: Combinatorics Elementary number theory Sequences and sets

Published online by Cambridge University Press:  03 October 2024

Ji-Cai Liu
Ji-Cai Liu*
Affiliation:
Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang, 325035, PR China
*
Email: jcliu2016@gmail.com
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Abstract

Let Mn and Tn denote the nth Motzkin number and the nth central trinomial coefficient respectively. We prove that for any prime $p\ge 5$,

\begin{equation*}\begin{aligned}\\[-22pt]&\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2},\\[-6pt]\end{aligned}\end{equation*}

where $\left(-\right)$ is the Legendre symbol. These results confirm three supercongruences conjectured by Z.-W. Sun in 2010.

Keywords

Motzkin numbersCatalan numberscentral trinomial coefficientssupercongruencesLegendre symbol

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 05A19: Combinatorial identities, bijective combinatorics 11B65: Binomial coefficients; factorials; $q$-identities

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 4 , November 2024 , pp. 1060 - 1084
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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References

Chen, W. Y. C., Hou, Q. -H. and Mu, Y. -P., A telescoping method for double summations, J. Comput. Appl. Math. 196(2) (2006), 553–566.CrossRefGoogle Scholar
Donaghey, R., and Shapiro, L. W., Motzkin numbers, J. Comb. Theory, Ser. A 23(3) (1977), 291–301.CrossRefGoogle Scholar
Gould, H. W., Combinatorial Identities, A standardized set of tables listing 500 binomial coefficient summations, Morgantown, WV, 1972, .CrossRefGoogle Scholar
Granville, A., The square of the Fermat quotient, Integers 4 (2004), .Google Scholar
Lehmer, E., On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. 39(2) (1938), 350–360.CrossRefGoogle Scholar
Motzkin, T., Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products, Bull. Am. Math. Soc. 54(4) (1948), 352–360.CrossRefGoogle Scholar
Pan, H., and Sun, Z. -W., A combinatorial identity with application to Catalan numbers, Discrete Math. 306(16) (2006), 1921–1940.CrossRefGoogle Scholar
Riordan, J., Combinatorial Identities, John Wiley & Sons, Inc., New York, London, Sydney, 1968, .Google Scholar
Schneider, C., Symbolic summation assists combinatorics, Sém. Lothar. Combin. 56 (2007), .Google Scholar
Sloane, N.J.A., The On-Line Encyclopedia of Integer Sequences, http://oeis.org.Google Scholar
Stanley, R. P., Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, .Google Scholar
Sun, Z. -W., Congruences involving generalized central trinomial coefficients, Sci. China Math. 57(7) (2014), 1375–1400.CrossRefGoogle Scholar
Sun, Z. -W., Congruences involving $g_n(x)=\sum_{k=0}^n{n\choose k}^2{2k \choose k}x^k$. Ramanujan J., 40(3) (2016), .Google Scholar
Sun, Z. -W., On Motzkin numbers and central trinomial coefficients, Adv. in Appl. Math. 136 (2022), Art. 102319.CrossRefGoogle Scholar
Sun, Z. -W. and Tauraso, R., New congruences for central binomial coefficients, Adv. in Appl. Math. 45(1) (2010), 125–148.CrossRefGoogle Scholar
Sun, Z. -W. and Tauraso, R., On some new congruences for binomial coefficients, Int. J. Number Theory 7(3) (2011), 645–662.CrossRefGoogle Scholar
Tauraso, R., More congruences for central binomial sums with Fibonacci and Lucas numbers, J. Integer Seq. 26(8) (2023), Art. 23.8.3.Google Scholar
Von Szily, K., Über die Quadratsummen der Binomialcoefficienten, Ungar. Ber. 12 (1894), 84–91.Google Scholar
Wolstenholme, J., On certain properties of prime numbers, Q. J. Pure Appl. Math. 5 (1862), 35–39.Google Scholar