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A167015 Triangle T, read by rows, where T(n,k) = [T^n](n-k-1,0); i.e., where row n of T equals the initial n terms of column 0 in matrix power T^n, reversed and with an appended '1', for n>0, with T(0,0)=1. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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.
LINKS
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
Sequence in context: A157109 A185814 A174553 * A124522 A016540 A132620
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 27 2009
EXTENSIONS
Edited by Paul D. Hanna, Oct 30 2009
STATUS
approved

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Last modified May 23 22:02 EDT 2024. Contains 372765 sequences. (Running on oeis4.)