Congruences involving Delannoy numbers and Schröder numbers

2024/10/23 by Jia, Chen-Bo, Huang, Jia-Qing · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.17522

Abstract

The central Delannoy numbers D_n=∑_k=0ⁿ\binomnk\binomn+kk and the little Schröder number s_n=∑_k=1ⁿ(1)/(n)\binomnk\binomnk-12^n-k are important quantities. In this paper, we confirm (2)/(3n(n+1))∑_k=1ⁿ (-1)^n-kk²D_kD_k-1 and \frac 1n∑_k=1ⁿ (-1)^n-k(4k²+2k-1)D_k-1s_kare positive odd integers for all n=1,2,3,⋯. We also show that for any prime number p>3, ∑_k=1^p-1 (-1)^kk²D_kD_k-1 ≡ -\frac56p \pmodp² and ∑_k=1^p (-1)^k(4k²+2k-1)D_k-1s_k ≡ -4p \pmodp². Moreover, define s_n(x)=∑_k=1ⁿ(1)/(n)\binomnk\binomnk-1x^k-1(x+1)^n-k, for any n∈ℤ⁺ is even we have (4)/(n(n+1)(n+2)(1+2x)³)∑_k=1ⁿk(k+1)(k+2)s_k(x)s_k+1(x)∈ℤ[x].

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