Log-convexity of Aigner–Catalan–Riordan numbers

Abstract Let T = [ t n , k ] n , k ≥ 0 be an infinite lower triangular matrix defined by t 0 , 0 = 1 , t n + 1 , 0 = ∑ j = 0 n z j t n , j , t n + 1 , k + 1 = ∑ j = k n a j , k t n , j for n , k ≥ 0 , where all z j , a j , k are nonnegative and a j , k = 0 unless j ≥ k ≥ 0 . We show that the sequence ( t n , 0 ) n ≥ 0 is log-convex if the coefficient matrix [ ζ , A ] is TP 2 , where ζ = [ z 0 , z 1 , z 2 , … ] ′ and A = [ a i , j ] i , j ≥ 0 . This gives a unified proof of the log-convexity of many well-known combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the Schroder numbers, the Bell numbers, and so on.