Cooper's one-term supercongruences for the sporadic sequences s_7 and s_{18}

Let

$$s_7(n)=\sum_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}\binom{2k}{n}$$

and

$$s_{18}(n)=\sum_{k=0}^{\lfloor n/3\rfloor}(-1)^k\binom{n}{k}\binom{2k}{k}\binom{2(n-k)}{n-k}\left[\binom{2n-3k-1}{n}+\binom{2n-3k}{n}\right],$$

with $s_{18}(0)=1$. Cooper's conjecture. For every prime $p\geqslant3$,

$$s_7(mp)\equiv s_7(m)\pmod{p^3},$$

and, for every prime $p$,

$$s_{18}(mp)\equiv s_{18}(m)\pmod{p^2}.$$

These congruences are sporadic-sequence analogues of known congruences for Apéry numbers. The paper proves the $s_7$ congruence for $p\geqslant5$ and the $s_{18}$ congruence for all primes; the $s_7$ case for $p=3$ remains open.