A104445 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT_UP(T) = T^2 - T + I, or, equivalently: T(n+1,k+1) = [T^2](n,k) - T(n,k) + [T^0](n,k) for n>=k>=0, with T(0,0)=1.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 9, 9, 4, 1, 1, 1, 24, 30, 16, 5, 1, 1, 1, 77, 115, 70, 25, 6, 1, 1, 1, 295, 510, 344, 135, 36, 7, 1, 1, 1, 1329, 2602, 1908, 805, 231, 49, 8, 1, 1, 1, 6934, 15133, 11904, 5325, 1616, 364, 64, 9, 1, 1, 1, 41351, 99367, 83028, 39001

Offset: 0

Author

Paul D. Hanna, Mar 07 2005

Keywords

• nonn
• tabl

Comments

Surprisingly, SHIFT_UP(T) = A091351, or T(n+1,k) = A091351(n,k) for n>=k>=0, where column k of A091351 equals column 0 of A091351^(k+1) for k>=0.

Examples

			Rows begin:
1;
1,1;
1,1,1;
1,2,1,1;
1,4,3,1,1;
1,9,9,4,1,1;
1,24,30,16,5,1,1;
1,77,115,70,25,6,1,1;
1,295,510,344,135,36,7,1,1;
1,1329,2602,1908,805,231,49,8,1,1;
1,6934,15133,11904,5325,1616,364,64,9,1,1; ...
		

Crossrefs

Cf. A091351, A104446 (matrix square); columns form: A091352, A091353, A091354.

Programs

PARI

T(n,k)=if(n
				

Formula

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