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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..63.

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

Example

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.

Sequence in context: A055248 A103316 A332389 * A105851 A106195 A247023

Adjacent sequences: A140066 A140067 A140068 * A140070 A140071 A140072

Keyword

nonn,tabl

Author

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

Status

approved