|
EXAMPLE
|
Triangle begins:
1;
1, 1;
1, 1, 1;
6, 3, 1, 1;
80, 25, 5, 1, 1;
1666, 378, 56, 7, 1, 1;
47232, 8460, 1020, 99, 9, 1, 1;
1694704, 252087, 26015, 2134, 154, 11, 1, 1;
73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
GENERATE T FROM ODD MATRIX POWERS OF T.
Matrix cube, T^3, begins:
1;
3, 1;
6, 3, 1; <-- row 3 of T
31, 12, 3, 1;
357, 100, 18, 3, 1;
6786, 1455, 205, 24, 3, 1; ...
where row 3 of T = row 2 of T^3 with appended '1'.
Matrix fifth power, T^5, begins:
1;
5, 1;
15, 5, 1;
80, 25, 5, 1; <-- row 4 of T
855, 215, 35, 5, 1;
15171, 3065, 410, 45, 5, 1; ...
where row 4 of T = row 3 of T^5 with appended '1'.
Matrix seventh power, T^7, begins:
1;
7, 1;
28, 7, 1;
161, 42, 7, 1;
1666, 378, 56, 7, 1; <-- row 5 of T
28119, 5348, 679, 70, 7, 1; ...
where row 5 of T = row 4 of T^7 with appended '1'.
ALTERNATE GENERATING METHOD.
Row 4: start with a '1' followed by 4 zeros;
take partial sums and append 2 zeros; then
take partial sums thrice more:
(1), 0, 0, 0, 0;
1, 1, 1, 1, (1), 0, 0;
1, 2, 3, 4, 5, 5, (5);
1, 3, 6, 10, 15, 20, (25);
1, 4, 10, 20, 35, 55, (80);
the final non-zero terms forms row 4: [80, 25, 5, 1, 1].
Row 5: start with a '1' followed by 6 zeros;
take partial sums and append 4 zeros;
take partial sums and append 2 zeros; then
take partial sums thrice more:
(1), 0, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, (1), 0, 0, 0, 0;
1, 2, 3, 4, 5, 6, 7, 7, 7, 7, (7), 0, 0;
1, 3, 6, 10, 15, 21, 28, 35, 42, 49, 56, 56, (56);
1, 4, 10, 20, 35, 56, 84, 119, 161, 210, 266, 322, (378);
1, 5, 15, 35, 70, 126, 210, 329, 490, 700, 966, 1288, (1666);
the final non-zero terms forms row 5: [1666, 378, 56, 7, 1, 1].
Continuing in this way produces all the rows of this triangle.
|
