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 trial and error puzzle, we can propose as decomposition:

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

Where:

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

In the last part of the proof, involving binomial coefficients, we have used: (See ref [1] and [2])

1)

2)

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 (III).nb

**References:**

[1]-Ronald L. Graham, Donald E. Knuth, and Oren Patashnik (Reading, Massachusetts: Addison-Wesley, 1994 – Concrete Mathematics – Identity (5.32) in Table 169.

[2]-Matthew Hubbard and Tom Roby – Pascal’s Triangle From Top To Bottom – the binomial coefficient website– Catalog #: 3100004.

[2]-Matthew Hubbard and Tom Roby – Pascal’s Triangle From Top To Bottom – the binomial coefficient website– Catalog #: 3100004.

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