New congruences for sums involving Apery numbers or central Delannoy numbers
arXiv:1008.2894
Abstract
The Apéry numbers and central Delannoy numbers are defined by 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 0\pmod n, where , , and . For , we further show that \sum_{k=0}^{n-1}(2k+1)^{3}A_k &\equiv 0\pmod{n^3}, \quad \sum_{k=0}^{p-1}(2k+1)^{3}A_k &\equiv p^3 \pmod{2p^6}, where is a prime. The following congruence \sum_{k=0}^{n-1} {n+k\choose k}^2{n-1\choose k}^2 \equiv 0 \pmod{n} plays an important role in our proof.
14 pages, to appear in International Journal of Number Theory