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- W2080285643 abstract "The Ap'ery numbers $A_n$ and central Delannoy numbers $D_n$ are defined by $$A_n=sum_{k=0}^{n}{n+kchoose 2k}^2{2kchoose k}^2, quad D_n=sum_{k=0}^{n}{n+kchoose 2k}{2kchoose k}. $$ Motivated by some recent work of Z.-W. Sun, we prove the following congruences: sum_{k=0}^{n-1}(2k+1)^{2r+1}A_k &equiv sum_{k=0}^{n-1}varepsilon^k (2k+1)^{2r+1}D_k equiv 0pmod n, where $ngeqslant 1$, $rgeqslant 0$, and $varepsilon=pm1$. For $r=1$, we further show that sum_{k=0}^{n-1}(2k+1)^{3}A_k &equiv 0pmod{n^3}, quad sum_{k=0}^{p-1}(2k+1)^{3}A_k &equiv p^3 pmod{2p^6}, where $p>3$ is a prime. The following congruence sum_{k=0}^{n-1} {n+kchoose k}^2{n-1choose k}^2 equiv 0 pmod{n} plays an important role in our proof." @default.
- W2080285643 created "2016-06-24" @default.
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- W2080285643 date "2010-08-17" @default.
- W2080285643 modified "2023-10-16" @default.
- W2080285643 title "New congruences for sums involving Apery numbers or central Delannoy numbers" @default.
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- W2080285643 doi "https://doi.org/10.48550/arxiv.1008.2894" @default.
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