A160185 Triangle read by rows, (1 / ((-1)*A129184 * A007318 + I)) - I, I = Identity matrix.

1, 2, 1, 5, 3, 1, 15, 9, 4, 1, 52, 31, 14, 5, 1, 203, 121, 54, 20, 6, 1, 877, 523, 233, 85, 27, 7, 1, 4140, 2469, 1101, 400, 125, 35, 8, 1, 21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1

Offset: 0

Author

Gary W. Adamson, May 03 2009

Keywords

• nonn
• tabl

Comments

Inverse binomial transform of the triangle shifts to left (= adding I as right border, I = Identity matrix); resulting in reversed rows of A121207.

Left border = Bell numbers, A000110 = eigensequence of Pascal's triangle.

Successive columns from left to right = eigensequences of Pascal's triangle deleting columns one at a time.

Row sums of the triangle = A060719: (1, 3, 9, 29, 103, ...). - Gary W. Adamson, May 20 2013

From Gary W. Adamson, Jul 18 2019: (Start)

Rows are eigensequences of triangles exemplified by the following arrangement of binomial sequences. Example: row 5 is (1, 5, 14, 31, 52, 0, 0, 0, ...), the eigensequence of:

1;

4, 1;

6, 3, 1;

4, 3, 2, 1;

1, 1, 1, 1, 1;

... and the rest zeros.

Similarly, the production matrix for (1, 6, 20, 54, 121, 203, 0, 0, 0, ...) is:

1;

5, 1;

10, 4, 1;

10, 6, 3, 1;

5, 4, 3, 2, 1;

1, 1, 1, 1, 1, 1;

... and the rest zeros. (End)

Examples

			First few rows of the triangle:
       1;
       2,     1;
       5,     3,     1;
      15,     9,     4,     1;
      52,    31,    14,     5,    1;
     203,   121,    54,    20,    6,   1;
     877,   523,   233,    85,   27,   7,   1;
    4140,  2469,  1101,   400,  125,  35,   8,  1;
   21147, 12611,  5625,  2046,  635, 175,  44,  9,  1;
  115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1;
  ...
		

Crossrefs

Cf. A121207, A124496, A186020.

Cf. A060719.

Formula

Triangle read by rows, 1 / ((-1)*A129184 * A051731 + I), I = Identity matrix.

Equals reversal by rows of triangle A121207, then delete right border. A121207 begins: 1; 1, 1; 1, 1, 2 1, 1, 3, 5; ...

Extensions

Corrected by Alois P. Heinz, Apr 18 2013