On the binomial transforms of Ap\'ery-like sequences

In the proof of the irrationality of (3) and (2), Ap\'ery defined two integer sequences through 3-term recurrences, which are known as the famous Ap\'ery numbers. Zagier, Almkvist--Zudilin and Cooper successively introduced the other 13 sporadic sequences through variants of Ap\'ery's 3-term recurrences. All of the 15 sporadic sequences are called Ap\'ery-like sequences. Motivated by Gessel's congruences mod 24 for the Ap\'ery numbers, we investigate the congruences in the form uₙ ⁿ N_~ (Z, N_ N^+) for all of the 15 Ap\'ery-like sequences \uₙ\₍ ₀. Let N_ be the largest positive integer such that uₙ ⁿ N_ for all non-negative integers n. We determine the values of \N_{| Z\} for all of the 15 Ap\'ery-like sequences \uₙ\₍ ₀. The binomial transforms of Ap\'ery-like sequences provide us a unified approach to this type of congruences for Ap\'ery-like sequences.