A185620 Triangular matrix T that satisfies: T^3 - T^2 + I = SHIFT_LEFT(T), as read by rows.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 10, 5, 1, 1, 1, 42, 27, 7, 1, 1, 1, 226, 173, 52, 9, 1, 1, 1, 1525, 1330, 442, 85, 11, 1, 1, 1, 12555, 12134, 4345, 897, 126, 13, 1, 1, 1, 123098, 129359, 49114, 10687, 1586, 175, 15, 1, 1, 1, 1408656, 1587501, 632104, 143335, 22156, 2557

Offset: 0

Author

Paul D. Hanna, Feb 01 2011

Keywords

• nonn
• tabl

Examples

			Triangle T begins:
1;
1, 1;
1, 1, 1;
1, 3, 1, 1;
1, 10, 5, 1, 1;
1, 42, 27, 7, 1, 1;
1, 226, 173, 52, 9, 1, 1;
1, 1525, 1330, 442, 85, 11, 1, 1;
1, 12555, 12134, 4345, 897, 126, 13, 1, 1;
1, 123098, 129359, 49114, 10687, 1586, 175, 15, 1, 1;
1, 1408656, 1587501, 632104, 143335, 22156, 2557, 232, 17, 1, 1;
1, 18499835, 22127494, 9167575, 2149761, 343091, 40936, 3858, 297, 19, 1, 1; ...
Matrix square T^2 begins:
1;
2, 1;
3, 2, 1;
6, 7, 2, 1;
18, 28, 11, 2, 1;
79, 142, 66, 15, 2, 1;
463, 913, 470, 120, 19, 2, 1;
3396, 7244, 3997, 1098, 190, 23, 2, 1;
...
Matrix cube T^3 begins:
1;
3, 1;
6, 3, 1;
16, 12, 3, 1;
60, 55, 18, 3, 1;
305, 315, 118, 24, 3, 1;
1988, 2243, 912, 205, 30, 3, 1;
15951, 19378, 8342, 1995, 316, 36, 3, 1;
...
Thus T^3 - T^2 + I begins:
1;
1, 1;
3, 1, 1;
10, 5, 1, 1;
42, 27, 7, 1, 1;
226, 173, 52, 9, 1, 1;
1525, 1330, 442, 85, 11, 1, 1;
12555, 12134, 4345, 897, 126, 13, 1, 1;
...
which equals T shifted left one column.
...
ALTERNATE GENERATING FORMULA.
Let U equal T shifted up one diagonal:
1;
1, 1;
1, 3, 1;
1, 10, 5, 1;
1, 42, 27, 7, 1;
1, 226, 173, 52, 9, 1;
1, 1525, 1330, 442, 85, 11, 1;
1, 12555, 12134, 4345, 897, 126, 13, 1;
...
then U*T^2 begins:
1;
3, 1;
10, 5, 1;
42, 27, 7, 1;
226, 173, 52, 9, 1;
1525, 1330, 442, 85, 11, 1;
12555, 12134, 4345, 897, 126, 13, 1;
...
which equals U shifted left one column.
		

Crossrefs

Cf. columns: A185621, A185622, A185623; A185624 (T^2), A185628 (T^3).

Cf. variants: A104445, A185641.

Programs

PARI

{T(n, k)=local(A=Mat(1), B); for(m=1, n, B=A^3-A^2+A^0;
A=matrix(m+1, m+1); for(i=1, m+1, for(j=1, i, if(i<2|j==i, A[i, j]=1,
if(j==1, A[i, j]=1, A[i, j]=B[i-1, j-1]))))); return(A[n+1, k+1])}

Formula

Recurrence: T(n+1,k+1) = [T^3](n,k) - [T^2](n,k) + [T^0](n,k) for n>=k>=0, with T(n,0)=1 for n>=0.

Let U equal T shifted up one diagonal; then U*T^2 equals U shifted left one column.