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 one step to get matrix .
Add a diagonal of ones to the main diagonal 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 Catalan numbers in all columns.
Keywords: Catalan numbers, INVERT transform, matrix inverse.