Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k weak ascents (1<=k<=n-1 for n>=2; k=1 for n=1). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.

A114691

Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k weak ascents (1<=k<=n-1 for n>=2; k=1 for n=1). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.