Mats Granvik mats.granvik(AT)abo.fi and Gary W. Adamson INVERT transform
Consider the Pascal triangle:
Shift down the elements one step and add a diagonal of ones in the main diagonal.
Change the sign of all the elements below the main diagonal. is found as a comment here: Oeis table A121207.
Calculate the matrix inverse and you will get:
As we see we have the Bell numbers in the first column. Oeis, Bell numbers. The INVERT transform of any number triangle is equivalent to the steps above, here with the INVERT transform of the Pascal triangle as example.
Repeating the procedure or algorithm (above), infinitely many times, will produce the Catalan numbers as a convergent Oeis Catalan numbers in all columns, regardless of starting triangle (or starting sequence). That is you input matrix into the first step instead of matrix .
Another way to calculate the first column in matrix A is by taking matrix powers of this matrix M. Here the elements in the main diagonal have been deleted except for the first element which is equal to 1. Taking matrix powers is simply matrix multiplication repeated, , and so on.
As promised we have calculated the Bell numbers in the first column.
Yet another way to calculate the first column in matrix , is to calculate permanents of a modified version of matrix . Here the element in the lower right corner has been swapped with the element in the lower right corner.
i.e. the last element in the first column of matrix .
Similarly for submatrices:
i.e. the second last element in the first column of matrix .