On Delannoy numbers and Schröder numbers

Abstract

The n-th Delannoy number and the n-th Schröder number given by $D_n={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)$ and $S_n={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)/(k+1)$ respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that $$\sum _{k=1}^{p-1}{D}_k/k^2=2(-1/p)E_{p-3}(modp)$$ and $$\sum _{k=1}^{p-1}{S}_k/m^k=(m^2-6m+1)/(2m)\ast (1-((m^2-6m+1)/p)(modp),$$ where (-) is the Legendre symbol, E_0,E_1,E_2,... are Euler numbers and m is any integer not divisible by p. We also conjecture that ${\sum }_{k=1}^{p-1}{D}_k^2/k^2=-2q_p(2{)}^2(modp)$, where $q_p(2)=(2^{p-1}-1)/p$.

Cited by in corpus (6)

• Open Conjectures on Congruences
• On congruences related to central binomial coefficients
• A determinantal expression and a recursive relation of the Delannoy numbers
• Congruences involving generalized central trinomial coefficients
• An Extension of a Congruence by Kohnen
• A search for primes $p$ such that Euler number $E_{p-3}$ is divisible by $p$