On congruences for sums of Apéry-like numbers and harmonic numbers

In 2006, Almkvist and Zudilin introduced the first-kind Apéry-like sequence Tn=∑k=0n(nk)2(2kn)2. By combining binomial identities with congruences for harmonic numbers, we establish congruences modulo powers of an odd prime for sums involving the numbers Tn. In particular, for α∈{1,3}, we evaluate ∑k=0p−1(2k+1)α4kTkHk modulo p4, where Hk=∑j=1k1/j. We also prove related congruences for a three-parameter family of Apéry-like polynomials.