OFFSET
0,4
COMMENTS
Although only the right portion of this triangle is shown here, by definition this triangular matrix can be extended infinitely to the left as well.
EXAMPLE
Triangle T begins:
1;
1,1;
2,1,1;
9,3,1,1;
70,14,4,1,1;
755,105,20,5,1,1;
10166,1080,149,27,6,1,1;
161350,13916,1491,203,35,7,1,1;
2917524,212634,18612,2002,268,44,8,1,1;
58811631,3723198,275856,24429,2628,345,54,9,1,1;
1302452122,73047825,4699180,353300,31562,3385,435,65,10,1,1;
31362843270,1581256303,89993827,5877619,447568,40227,4290,539,77,11,1,1;
814897356483,37350588290,1907644760,110140398,7295576,561607,50662,5361,658,90,12,1,1;
22712570157056,954796686233,44253266889,2289730547,134056234,8995558,698737,63128,6617,793,104,13,1,1;
675859219349848,26246234745486,1113845999245,52215711142,2736921663,162397676,11026087,862680,77910,8078,945,119,14,1,1; ...
in which row n is formed from column 0 of T^n.
Matrix square T^2 begins:
1;
2, 1;
5, 2, 1;
23, 7, 2, 1;
171, 35, 9, 2, 1;
1770, 259, 49, 11, 2, 1;
23128, 2579, 369, 65, 13, 2, 1;
359326, 32237, 3614, 503, 83, 15, 2, 1; ...
where column 0 of T^2 forms row 2 of T: [2,1,1].
Matrix cube T^3 begins:
1;
3, 1;
9, 3, 1;
43, 12, 3, 1;
312, 64, 15, 3, 1;
3121, 474, 88, 18, 3, 1;
39616, 4615, 675, 115, 21, 3, 1;
602135, 56173, 6538, 918, 145, 24, 3, 1; ...
where column 0 of T^3 forms row 3 of T: [9,3,1,1].
Matrix 4th power T^4 begins:
1;
4, 1;
14, 4, 1;
70, 18, 4, 1;
503, 102, 22, 4, 1;
4898, 763, 138, 26, 4, 1;
60517, 7324, 1083, 178, 30, 4, 1;
899764, 87171, 10454, 1467, 222, 34, 4, 1; ...
where column 0 of T^4 forms row 4 of T: [70,14,4,1,1].
Matrix 5th power T^5 begins:
1;
5, 1;
20, 5, 1;
105, 25, 5, 1;
755, 150, 30, 5, 1;
7206, 1140, 200, 35, 5, 1;
86895, 10861, 1610, 255, 40, 5, 1;
1264270, 126940, 15576, 2170, 315, 45, 5, 1; ...
where column 0 of T^5 forms row 5 of T: [755,105,20,5,1,1].
Matrix 6th power T^6 begins:
1;
6, 1;
27, 6, 1;
149, 33, 6, 1;
1080, 209, 39, 6, 1;
10166, 1620, 275, 45, 6, 1;
120012, 15401, 2274, 347, 51, 6, 1;
1710097, 177477, 22142, 3048, 425, 57, 6, 1; ...
where column 0 of T^6 forms row 6 of T: [10166,1080,149,27,6,1,1].
PROG
(PARI) {T(n, k)=local(M, N); if(n==k||n==k+1, 1, if(n==k+2, k+2, N=matrix(n, n, r, c, if(r>=c, T(r-1, c-1))); M=matrix(n+1, n+1, r, c, if(r>=c, if(r<=n, N[r, c], (N^n)[n-k, 1]))); M[n+1, k+1]))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 27 2009
EXTENSIONS
Edited by Paul D. Hanna, Oct 30 2009
STATUS
approved