Let be $latex U_n$ an square matrix with elements:
$latex \displaystyle u_{i,j}=\binom{j}{i}$
$latex U_n$ is an upper triangular matrix with all its diagonal elements equal to $latex 1$, and similar to Pascal's triangle.(That I prefer to name Tartaglia's Triangle)
Example:
$latex U_{10}= \left( \begin{array}{cccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 0 & 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & 45\\ 0 & 0 & 1 & 4 & 10 & 20 & 35 & 56 & 84 & 120 \\ 0 & 0 & 0 & 1 & 5 & 15 & 35 & 70 & 126 & 210 \\ 0 & 0 & 0 & 0 & 1 & 6 & 21 & 56 & 126 & 252 \\ 0 & 0 & 0 & 0 & 0 & 1 & 7 & 28 & 84 & 210 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 8 & 36 & 120 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 9 & 45 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 10 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right)$
And, of course:
$latex |U_{n}|=1$
If we multiply this matrix by its transposed one, then we get a symmetrical matrix with:
$latex |A_{n}|=|U_{n}^{T}*U_{n}|=|U_{n}|^2=1$
Example:
$latex A_{10}=U_{10}^{T}*U_{10}$
$latex \displaystyle A_{10}=\\ \\ \left( \begin{array}{cccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & 54 & 65 \\ 3 & 9 & 19 & 34 & 55 & 83 & 119 & 164 & 219 & 285 \\ 4 & 14 & 34 & 69 & 125 & 209 & 329 & 494 & 714 & 1000 \\ 5 & 20 & 55 & 125 & 251 & 461 & 791 & 1286 & 2001 & 3002 \\ 6 & 27 & 83 & 209 & 461 & 923 & 1715 & 3002 & 5004 & 8007 \\ 7 & 35 & 119 & 329 & 791 & 1715 & 3431 & 6434 & 11439 & 19447 \\ 8 & 44 & 164 & 494 & 1286 & 3002 & 6434 & 12869 & 24309 & 43757 \\ 9 & 54 & 219 & 714 & 2001 & 5004 & 11439 & 24309 & 48619 & 92377 \\ 10 & 65 & 285 & 1000 & 3002 & 8007 & 19447 & 43757 & 92377 & 184755 \end{array}\right)$
So we have constructed a new matrix with determinant equal to one:
$latex \displaystyle a_{i,j}=\sum_{r=1}^n{u_{i,r}^{*} \cdot u_{r,j}}=\sum_{r=1}^{min(i,j)}{u_{r,i} \cdot u_{r,j}}$
$latex \displaystyle a_{i,j}=\sum_{r=1}^{min(i,j)}{\binom{i}{r} \binom{j}{r}}=-1+\sum_{r=0}^{i}{\binom{i}{r} \binom{j}{r}}=\binom{i+j}{i}-1=\binom{i+j}{j}-1$
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.
