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

0,8

Links

Table of n, a(n) for n=0..61.

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.

Example

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.

Prog

(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])}

Crossrefs

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

Cf. variants: A104445, A185641.

Sequence in context: A307847 A080214 A263383 * A096066 A294746 A326180

Adjacent sequences: A185617 A185618 A185619 * A185621 A185622 A185623

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 01 2011

Status

approved