If is an square matrix with elements:

Then:

*Proof:*The matrix can be decomposed as the product of a lower triangular matrix, , and an upper triangular matrix, :

*Example:*

After some math “plumbing”, we can propose as decomposition:

If this decomposition is correct then the matrix product should be :

Where:

, can be simplified, changing the binomial coefficient to the gamma function, as:

This last expression can be simplified, with the help of:

, the Vandermonde´s convolution (see reference [2]), and the absortion formula (see reference [3]), to:

Now we can calculate easily , because the LU-decomposition used is the so called Doolittle decomposition, where the matrix has all ones on its diagonal.

**Archives:**[a]-092609-Binomial Matrix (II).nb

**References:**

[1]-John H. Mathews and Kurtis Fink, 2004 – Module for PA = LU Factorization with Pivoting

[2]-Matthew Hubbard and Tom Roby – Pascal’s Triangle From Top To Bottom – the binomial coefficient website– Catalog #: 3100003.

[3]-Matthew Hubbard and Tom Roby – Pascal’s Triangle From Top To Bottom – the binomial coefficient website– Catalog #: 2400002.

[2]-Matthew Hubbard and Tom Roby – Pascal’s Triangle From Top To Bottom – the binomial coefficient website– Catalog #: 3100003.

[3]-Matthew Hubbard and Tom Roby – Pascal’s Triangle From Top To Bottom – the binomial coefficient website– Catalog #: 2400002.

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