Scholar - SciOpen

Let ${S_{n}}$ be the Apéry-like sequence given by $S_{n} = \sum_{k = 0}^{n} (\binom{n}{k}) (\binom{2 k}{k}) (\binom{2 n - 2 k}{n - k})$ . We show that for any odd prime $p$ , $\sum_{n = 1}^{p - 1} \frac{n S_{n}}{8^{n}} \equiv (1 - (- 1)^{\frac{p - 1}{2}}) p^{2} (mod p^{3})$ . Let ${Q_{n}}$ be the Apéry-like sequence given by $Q_{n} = \sum_{k = 0}^{n} (\binom{n}{k}) (- 8)^{n - k} \sum_{r = 0}^{k} {(\binom{k}{r})}^{3}$ . We establish many congruences concerning $Q_{n}$ . For an odd prime $p$ , we also deduce congruences for $\sum_{k = 0}^{p - 1} {(\binom{2 k}{k})}^{3} \frac{1}{64^{k}} (mod p^{3})$ , $\sum_{k = 0}^{p - 1} {(\binom{2 k}{k})}^{3} \frac{1}{64^{k} (k + 1)^{2}} (mod p^{2})$ and $\sum_{k = 0}^{p - 1} {(\binom{2 k}{k})}^{3} \frac{1}{64^{k} (2 k - 1)} (mod p)$ , and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.