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On some congruences involving Domb numbers and harmonic numbers. (English) Zbl 1433.11017

The authors prove three congruences involving Apéry-like numbers \(D_n\), \(S_n\) and \(T_n\) which have been conjectured by Sun. These numbers are defined by
\[ D_n = \sum_{k = 0}^n \binom{n}{k}^2 \binom{2k}{k}\binom{n}{k}\binom{2n-2k}{n-k},\] \[ S_n = \sum_{k = 0}^n \binom{n}{k} \binom{2k}{k}\binom{n}{k}\binom{2n-2k}{n-k},\] \[ T_n = \sum_{k = 0}^n \binom{n}{k}^2 \binom{2k}{n}^2. \]
The first conjecture proved is: Let \(p>3\) be a prime, then
\[ D_{p-1} \equiv 64^{p-1} - \frac{p^3}{6} B_{p-3} \pmod{p^4}. \]
Here \(B_n\) denotes as usual the Bernoulli numbers.
Moreover, they prove the following theorems for \(S_n\) and \(T_n\).
Theorem. Let \(p>3\) be a prime, then
\[ S_{p-1} \equiv (-1)^{\frac{p-1}{2}}32^{p-1} + p^2E_{p-3} \pmod{p^3}. \] Here \(E_n\) denote the Euler numbers.
Theorem. Let \(p>3\) be a prime, then
\[ T_{p-1} \equiv 16^{p-1} + \frac{p^3}{4} B_{p-3} \pmod{p^4}. \]

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11B75 Other combinatorial number theory
11A07 Congruences; primitive roots; residue systems
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