@article{Sun2022, 
author = {Zhi-Hong Sun},
title = {Supercongruences involving Apéry-like numbers and binomial coefficients},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {2},
pages = {2729-2781},
keywords = {congruence, binomial coefficient, Apéry-like number, Euler number, binary quadratic form},
url = {https://www.sciopen.com/article/10.3934/math.2022153},
doi = {10.3934/math.2022153},
abstract = {Let    {      S    n    } be the Apéry-like sequence given by        S    n    =      ∑          k      =      0        n              (              n      k              )                  (                      2        k            k              )                  (                      2        n        −        2        k                    n        −        k                    )      . We show that for any odd prime    p,        ∑          n      =      1              p      −      1                  n              S        n                    8      n            ≡    (  1  −  (  −  1      )                            p          −          1                2              )      p    2      (  mod              p      3        ). Let    {      Q    n    } be the Apéry-like sequence given by        Q    n    =      ∑          k      =      0        n              (              n      k              )        (  −  8      )          n      −      k            ∑          r      =      0        k                      (                    k        r                    )              3  . We establish many congruences concerning        Q    n  . For an odd prime    p, we also deduce congruences for        ∑          k      =      0              p      −      1                          (                              2          k                k                    )              3        1          64      k          (  mod              p      3        ),        ∑          k      =      0              p      −      1                          (                              2          k                k                    )              3        1                  64        k            (      k      +      1              )        2                (  mod              p      2        ) and        ∑          k      =      0              p      −      1                          (                              2          k                k                    )              3        1                  64        k            (      2      k      −      1      )          (  mod    p  ), and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.}
}