Symmetric n x n matrix of binomial coefficients.
1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 3, 6, 1, 1, 1, 1, 1, 2, 3, 4, 1, 3, 6, 10, 1, 4, 10, 20, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 1, 3, 6, 10, 15, 1, 4, 10, 20, 35, 1, 5, 15, 35, 70, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 1, 3, 6, 10, 15, 21, 1, 4, 10, 20, 35, 56, 1, 5, 15, 35, 70, 126, 1, 6, 21, 56, 126, 252
1
The binomial coefficients can be read in the antidiagonals of the matrices. The determinant of these matrices is 1. Their inverse, which is all integer, is given in S000062. The permanent is in S000063.
T. D. Noe, Plot of 30 rows
T. D. Noe, 30 Rows of numbers
Example: The third row is for the 3 x 3 matrix
1, 1, 1
1, 2, 3
1, 3, 6
(Mma) Table[m = Table[Binomial[i + j, i], {i, 0, n}, {j, 0, n}]; Flatten[m], {n, 0, 5}]
nonn,tabf,nice
T. D. Noe, May 27 2014