If we use yesterday´s idea, with little variations, we can create new expressions for matrices with elements related to binomial coefficients, for instance:
Let be an square matrix with elements:
is an upper triangular matrix with all its diagonal elements equal to , and similar to Pascal´s triangle.(That I prefer to name Tartaglia´s Triangle)
Example:
And, of course:
If we multiply this matrix by its transposed one, then we get a symmetrical matrix with:
Example:
So we have constructed a new matrix with determinant equal to one:
Note(i): For a proof on binomial identity see references [1] or [2]
OEIS Related Sequences:
Row/Column |
Sequence |
1 | A000027 |
2 | A000096 |
3 | A062748 |
4 | A063258 |
5 | A062988 |
6 | A124089 |
7 | A124090 |
8 | A165618 |
9 | A035927 |
10 | – |
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.