A Stern-type congruence for the Schroder numbers

TL;DR

This paper establishes a new congruence relation for Schröder numbers, revealing a specific modular pattern involving powers of two, which deepens understanding of their number-theoretic properties.

Contribution

The paper introduces a novel Stern-type congruence for Schröder numbers, expanding the theoretical framework of their modular behavior.

Findings

Proves a congruence relation for Schröder numbers modulo powers of two.

Shows that Schröder numbers follow a predictable pattern when indices are shifted by powers of two.

Enhances understanding of the number-theoretic properties of Schröder numbers.

Abstract

For the Schr\"oder number $$ S_n=\sum_{k=0}^n\binom{n}k\binom{n+k}k\frac1{k+1}, $$ we prove that $$ S_{n+2^\alpha}\equiv S_{n}+2^{\alpha+1}\pmod{2^{\alpha+2}}, $$ where $n\geq 1$ and $\alpha\geq 1$.