%I
%S 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,
%T 128,127,321,201,150,48,11,1,256,255,769,522,501,198,70,12,1,512,511,
%U 1793,1291,1524,699,338,82,14,1,1024,1023,4097,3084,4339,2223,1375,420
%N Triangle read by rows, nth row = (n1)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.
%C Sum of nth row terms = A001906(2n). Example: sum of 4th row terms = ( 8 + 7 + 5 + 1) = 21 = A001906(8).
%F Triangle read by rows, nth row = (n1)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.
%F 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
%e First few rows of the triangle are:
%e 1;
%e 2, 1;
%e 4, 3, 1;
%e 8, 7, 5, 1;
%e 16, 15, 17, 6, 1;
%e 32, 31, 49, 23, 8, 1;
%e 64, 63, 129, 72, 39, 9, 1;
%e 128, 127, 321, 201, 150, 48, 11, 1;
%e 256, 255, 769, 522, 501, 198, 70, 12, 1;
%e 512, 511, 1793, 1291, 1524, 699, 338, 82, 14, 1;
%e 1024, 1023, 4097, 3084, 4339, 2223, 1375, 420, 110, 15, 1;
%e ...
%Y Cf. A140068.
%Y Cf. A135225.
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_ and _Roger L. Bagula_, May 04 2008
