In this work, we establish some characteristics for a sequence, Aα(n,k), including recurrence relations, generating function and inversion formula, etc. Based on the sequence, we derive, by means of the generating function approach, some transformation formulas concerning certain combinatorial numbers named after Lah, Stirling, harmonic, Cauchy and Catalan, as well as several closed finite sums. In addition, the relationship between Aα(n,k) and r-Whitney–Lah numbers is established, and some formulas for the r-Whitney–Lah numbers are obtained.