An extension of Gauss congruences for Ap\'ery numbers

Osburn, Sahu and Straub introduced the numbers: align* Aₙ^ (r, s, t) =₊=₀ⁿn kʳn+k kˢ2k nᵗ, 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 \Aₙ^{ (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).