A152406 Triangle T, read by rows, where column k of T = column 0 of matrix power T^(k+1) for k>0, with column 0 of T = column 1 of T^2 (shifted).

1, 1, 1, 4, 2, 1, 26, 10, 3, 1, 224, 74, 18, 4, 1, 2346, 698, 150, 28, 5, 1, 28516, 7838, 1546, 260, 40, 6, 1, 391042, 100850, 18642, 2916, 410, 54, 7, 1, 5936376, 1451454, 254690, 37712, 4980, 606, 70, 8, 1, 98435034, 22985130, 3861782, 547240, 68910

Offset

0,4

Links

Table of n, a(n) for n=0..49.

Formula

Column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.

Column k: T(n,k) = Sum_{j=0..n-k} T(n-k,j)*T(j+k-1,k-1) for n>=k>0.

Column 0: T(n,0) = Sum_{j=1..n} T(n,j)*T(j,1) for n>=0.

Example

Triangle T begins:
1;
1, 1;
4, 2, 1;
26, 10, 3, 1;
224, 74, 18, 4, 1;
2346, 698, 150, 28, 5, 1;
28516, 7838, 1546, 260, 40, 6, 1;
391042, 100850, 18642, 2916, 410, 54, 7, 1;
5936376, 1451454, 254690, 37712, 4980, 606, 70, 8, 1;
98435034, 22985130, 3861782, 547240, 68910, 7934, 854, 88, 9, 1;...
where column k of T = column 0 of T^(k+1) for k>0
and column 0 of T = column 1 of T^2 (shifted).
Amazingly, column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.
Matrix square T^2 begins:
1;
2, 1;
10, 4, 1;
74, 26, 6, 1;
698, 224, 48, 8, 1;
7838, 2346, 474, 76, 10, 1;
100850, 28516, 5492, 848, 110, 12, 1;
1451454, 391042, 72334, 10804, 1370, 150, 14, 1;...
where column 0 of T^2 = column 1 of T,
and column 2 of T^2 = column 1 of T^3.
Matrix cube T^3 begins:
1;
3, 1;
18, 6, 1;
150, 48, 9, 1;
1546, 474, 90, 12, 1;
18642, 5492, 1032, 144, 15, 1;
254690, 72334, 13362, 1884, 210, 18, 1;
3861782, 1060412, 192192, 27040, 3090, 288, 21, 1;...
where column 0 of T^3 = column 2 of T,
and column 3 of T^3 = column 2 of T^4.
Matrix power T^4 begins:
1;
4, 1;
28, 8, 1;
260, 76, 12, 1;
2916, 848, 144, 16, 1;
37712, 10804, 1884, 232, 20, 1;
547240, 153840, 27040, 3488, 340, 24, 1;
8751688, 2410328, 423240, 55840, 5780, 468, 28, 1;...
where column 0 of T^4 = column 3 of T,
and column 1 of T^4 = column 3 of T^2.

Prog

(PARI) T(n, k)=if(k>n || n<0, 0, if(k==n, 1, if(k==0, sum(j=1, n, T(n, j)*T(j, 1)), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ))

Crossrefs

Cf. columns: A152407, A152408, A152409, A152410.

Sequence in context: A111559 A224798 A239894 * A105623 A364870 A158835

Adjacent sequences: A152403 A152404 A152405 * A152407 A152408 A152409

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 05 2008

Status

approved