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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

Table of n, a(n) for n=0..56.

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

Cf. A167016, A167017, A167018, A167019; A167142, A167144, A167145.

Sequence in context: A157109 A185814 A174553 * A124522 A016540 A132620

Adjacent sequences:  A167012 A167013 A167014 * A167016 A167017 A167018

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 October 17 21:17 EDT 2021. Contains 348065 sequences. (Running on oeis4.)