A140069 Triangle read by rows, n-th row = (n-1)-th power of the matrix X * [1,0,0,0,...]; where X = an infinite lower triangular bidiagonal matrix with [2,1,2,1,2,1,...] and [1,1,1,...] in the subdiagonal.

1, 2, 1, 4, 3, 1, 8, 7, 5, 1, 16, 15, 17, 6, 1, 32, 31, 49, 23, 8, 1, 64, 63, 129, 72, 39, 9, 1, 128, 127, 321, 201, 150, 48, 11, 1, 256, 255, 769, 522, 501, 198, 70, 12, 1, 512, 511, 1793, 1291, 1524, 699, 338, 82, 14, 1, 1024, 1023, 4097, 3084, 4339, 2223, 1375, 420, 110, 15, 1

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, May 04 2008

Keywords

• nonn
• tabl

Comments

Sum of n-th row terms = A001906(2n). Example: sum of 4th row terms = ( 8 + 7 + 5 + 1) = 21 = A001906(8).

Examples

			First few rows of the triangle are:
  1;
  2, 1;
  4, 3, 1;
  8, 7, 5, 1;
  16, 15, 17, 6, 1;
  32, 31, 49, 23, 8, 1;
  64, 63, 129, 72, 39, 9, 1;
  128, 127, 321, 201, 150, 48, 11, 1;
  256, 255, 769, 522, 501, 198, 70, 12, 1;
  512, 511, 1793, 1291, 1524, 699, 338, 82, 14, 1;
  1024, 1023, 4097, 3084, 4339, 2223, 1375, 420, 110, 15, 1;
  ...
		

Crossrefs

Cf. A140068.

Cf. A135225.

Formula

Triangle read by rows, n-th row = (n-1)-th power of the matrix X * [1,0,0,0,...] where X = an infinite lower triangular matrix with [1,2,1,2,1,2,...] in the main diagonal and [1,1,1,...] in the subdiagonal, with rest zeros. Perform X * [1,0,0,0,...], X * result, etc; with the result of each operation generating successive rows of the triangle.

Binomial transform of A135225, as lower triangular matrices: a(n+1,k+1) = Sum_{j=0..n} binomial(n,j)*A135225(j,k). - Gary W. Adamson, Mar 01 2012