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EXAMPLE
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Triangle begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
194, 39, 6, 1, 1;
4114, 648, 76, 8, 1, 1;
118042, 15465, 1510, 125, 10, 1, 1;
4274612, 483240, 41121, 2908, 186, 12, 1, 1;
186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...
GENERATE T FROM EVEN MATRIX POWERS OF T.
Matrix square T^2 begins:
1;
2, 1; <-- row 2 of T
5, 2, 1;
34, 9, 2, 1;
453, 88, 13, 2, 1; ...
where row 2 of T = row 1 of T^2 with appended '1'.
Matrix fourth powers T^4 begins:
1;
4, 1;
14, 4, 1; <-- row 3 of T
96, 22, 4, 1;
1215, 220, 30, 4, 1; ...
where row 3 of T = row 2 of T^4 with appended '1'.
Matrix sixth power T^6 begins:
1;
6, 1;
27, 6, 1;
194, 39, 6, 1; <-- row 4 of T
2394, 404, 51, 6, 1; ...
where row 4 of T = row 3 of T^6 with appended '1'.
ALTERNATE GENERATING METHOD.
Start with [1,0,0,0]; take partial sums and append 1 zero;
take partial sums twice more:
(1), 0, 0, 0;
1, 1, 1, (1), 0;
1, 2, 3, 4, (4);
1, 3, 6, 10, (14);
the final non-zero terms forms row 3: [14,4,1,1].
Start with [1,0,0,0,0,0]; take partial sums and append 3 zeros;
take partial sums and append 1 zero; take partial sums twice more:
(1), 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, (1), 0, 0, 0;
1, 2, 3, 4, 5, 6, 6, 6, (6), 0;
1, 3, 6, 10, 15, 21, 27, 33, 39, (39);
1, 4, 10, 20, 35, 56, 83, 116, 155, (194);
the final non-zero terms forms row 4: [194,39,6,1,1].
Continuing in this way produces all the rows of this triangle.
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