A partial Motzkin path of length n is a lattice path from(0,0)to(n,k),which run through the integer points,consisting of up steps U=(1,1),down steps D=(1,-1)and horizontal steps H=(1,0),and it never goes below the x-axis.The number of Motzkin paths from(0,0)to(n,0)is called the n-th Motzkin number.The generating function of Motzkin numbers and the representation of the Riordan matrix of the number of partial Motzkin paths are obtained by using the kernel method.Finally,the gen-erating functions of partial Motzkin paths with restricted height are given by using recurrence relations and the linear algebraic method.Some relevant examples are presented here.
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