An extension of Gauss congruences for Ap\'ery numbers

Osburn, Sahu and Straub introduced the numbers: {align*} A_n(r,s,t)=∑ₖ₌₀^n{n k}^r{n+k k}^s{2k n}^t, {align*} for non-negative integers $n,r,s,t$ with r≥ 2, which includes two kinds of Ap\'ery numbers and four kinds of Ap\'ery-like numbers as special cases, and showed that the numbers ₙ(r,s,t)≥ 0 satisfy the Gauss congruences of order $3$. We establish an extension of Osburn--Sahu--Straub congruence through Bernoulli numbers, which is one step deep congruence of the Gauss congruence for Aₙ(r,s,t).