I have here left out the indices of the elements for easier typing but hopefully this can still be understood.
Shift down the elements in matrix two steps to get matrix .
Add two diagonals of ones to the main diagonal (where row index = column index) and to the diagonal below (where row index = column index + 1) in matrix to get matrix .
Change the sign of all the elements below the main diagonal in matrix so that you get matrix .
Calculate the matrix inverse of matrix to get matrix
Iterate by replacing matrix with matrix .
Matrix should then converge to the Motzkin numbers in all columns.
Keywords: Matrix inversion, Motzkin numbers, iteration.