Journal of Integer Sequences, Vol. 29 (2026), Article 26.5.2

Congruence Properties for the Delannoy Triangle Coefficients


Chahinaze Djadi
Faculty of Mathematics
USTHB, ATN Laboratory
El Alia, Po. Box 32
Bab Ezzouar, 16111
Algiers
Algeria

Hacène Belbachir
Faculty of Mathematics
USTHB, RECITS Laboratory
El Alia, Po. Box 32
Bab Ezzouar, 16111
Algiers
Algeria

Abstract:

Let $\binom{n}{k}_{[2]}$ denote the entry in the $n$th row and $k$th column of the Delannoy triangle. In the present paper, we extend Wolstenholme's theorem to Delannoy triangle coefficients: for any prime $p\geq 5$, we determine $\binom{2p}{p}_{[2]}$, $\binom{2p-1}{p-1}_{[2]}$ and, more generally, $\binom{np-1}{p-1}_{[2]}$ modulo $p^3$ in terms of the Fermat quotient $q_p(2)$. We also establish, for any odd prime $p$, the congruence $\sum_{0\leq k\leq n\leq p-1}\binom{n}{k}_{[2]}\equiv \bigl(\frac{2}{p}\bigr)$ (mod $p$), where $\bigl(\frac{\cdot}{p}\bigr)$ is the Legendre symbol.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A001850 A008288 A027907 A050151.)

Received April 23 2025; April 24 2025; August 18 2026; August 25 2026. Published in Journal of Integer Sequences, September 21 2026.

Return to Journal of Integer Sequences home page