×

Proof of Sun’s conjectures on Schröder-like numbers. (English) Zbl 1428.11011

Summary: For any non-negative integer \(n\), define \(R_n\) and \(R_n(x)\) by
\[ R_n = \sum_{k=0}^n \binom{n+k}{2k}\binom{2k}{k}\frac1{2k-1}\quad\text{and}\quad R_n(x) = \sum_{k=0}^n \binom{n+k}{2k}\binom{2k}{k}\frac{x^k}{2k-1}, \]
respectively. We mainly prove that for any positive integer \(n\) and odd prime \(p\), \[ \frac{3}{n} \sum_{k=0}^{n-1} R_k(x)^2 \in \mathbb Z[x],\]
\[ 3\,\sum_{k=0}^{p-1} R_k^2 \equiv (11+4(-1)^{(p+1)/2)}p \pmod{p^2}, \]
which were originally conjectured by Sun.

MSC:

11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions

Software:

SumTools
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H.-Q. Cao and H. Pan, A Stern-type congruence for the Schröder numbers, preprint (2015); arXiv:1512.06310v1.
[2] Guo, V. J. W., Some congruences involving powers of Legendre polynomials, Integral Transforms Spec. Funct.26 (2015) 660-666. · Zbl 1360.11007
[3] Guo, V. J. W. and Liu, J.-C., Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum, J. Number Theory156 (2015) 154-160. · Zbl 1395.11009
[4] Koepf, W., Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities (Friedrich Vieweg & Sohn, Braunschweig, 1998). · Zbl 0909.33001
[5] Liu, J.-C., A supercongruence involving Delannoy numbers and Schröder numbers, J. Number Theory168 (2016) 117-127. · Zbl 1396.11044
[6] Petkovšek, M., Wilf, H. S. and Zeilberger, D., \(A = B\) (A. K. Peters, Wellesley, MA, 1996). · Zbl 0848.05002
[7] Stanley, R. P., Hipparchus, Plutarch, Schröder and Hough, Amer. Math. Monthly104 (1997) 344-350. · Zbl 0873.01002
[8] Stanley, R. P., Enumerative Combinatorics, Vol. 2 (Cambridge University Press, Cambridge, 1999). · Zbl 0928.05001
[9] Sun, Z.-W., On Delannoy numbers and Schröder numbers, J. Number Theory131 (2011) 2387-2397. · Zbl 1280.11014
[10] Z.-W. Sun, Two new kinds of numbers and related divisibility results, preprint (2014); arXiv:1408.5381v8.
[11] Z.-W. Sun, Arithmetic properties of Delannoy numbers and Schröder numbers, preprint (2016); arXiv:1602.00574v3.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.