Log-convexity of Aigner–Catalan–Riordan numbers

Abstract

Let T=[tn,k]n,k≥0 be an infinite lower triangular matrix defined byt0,0=1,tn+1,0=∑j=0nzjtn,j,tn+1,k+1=∑j=knaj,ktn,j for n,k≥0, where all zj,aj,k are nonnegative and aj,k=0 unless j≥k≥0. We show that the sequence (tn,0)n≥0 is log-convex if the coefficient matrix [ζ,A] is TP2, where ζ=[z0,z1,z2,…]′ and A=[ai,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 Schröder numbers, the Bell numbers, and so on.