On Delannoy numbers and Schr\"oder numbers
By Zhi-Wei Sun. 2010
52 citations recorded by the source corpus.
The n-th Delannoy number and the n-th Schr\"oder number given by $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ and $S_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that $$\sum_{k=1}^{p-1}D_k/k^2=2(-1/p)E_{p-3} (mod p)$$ and $$\sum_{k=1}^{p-1}S_k/m^k=(m^2-6m+1)/(2m)*(1-((m^2-6m+1)/p) (mod p),$$ 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 $\sum_{k=1}^{p-1}D_k^2/k^2=-2q_p(2)^2 (mod p)$, where $q_p(2)=(2^{p-1}-1)/p$.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/9562de0eaab919420266280f2c21c3428cdab69d
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In Algebra and Number Theory
In Discrete Mathematics and Combinatorics
Citations in the corpus
- On sums of Ap\'ery polynomials and related congruences (2011, background): “ Wealsohave pX−1 k=0 (2k + 1)(−1)kDk ≡ p − 7 12 p4B p−3 (mod p 5) (1.18) and pX−1 k=0 (2k + 1)Dk ≡ p +2p2qp(2)− p3qp(2)2 (mod p4), (1.19) whereqp(2) denotestheFermatquotient(2p−1 −1)/p. Remark1.5. In [Su3] the author determined Pp−1 k=1Dk/k and Pp−1 k=1Dk/k 2 modulo an odd prime p. In the next secti”
- Conjectures involving arithmetical sequences (2012, background): “The reader may consult [26] for some congruences involving central Delannoy numbers and Schröder numbers.”