A140736 Triangle read by rows, X^n * [1,0,0,0,...]; where X = a tridiagonal matrix with (1,0,1,0,1,...) in the main diagonal and (1,1,1,...) in the sub- and subsubdiagonals.

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 5, 4, 6, 3, 1, 1, 1, 7, 6, 15, 10, 10, 4, 1, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1, 1, 1, 11, 10, 45, 36, 84, 56, 70, 35, 21, 6, 1, 1, 1, 13, 12, 66, 55, 165, 120, 210, 126, 126, 56, 28, 7, 1

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, May 25 2008

Keywords

• nonn
• tabf
• easy

Comments

A140737 = triangle with reversed terms by rows. - Gary W. Adamson, May 25 2008

T(n,k) is the element in column 1 of row k of the n-th power of the (2n+1)X(2n+1) tridiagonal matrix X with X(r,c) = 1 if (r=c and r odd) or r=c+1 or r=c+2. - R. J. Mathar, Nov 14 2023

Examples

			First few rows of the triangle are:
  1;
  1, 1,  1;
  1, 1,  3,  2,  1;
  1, 1,  5,  4,  6,  3,   1;
  1, 1,  7,  6, 15, 10,  10,   4,   1;
  1, 1,  9,  8, 28, 21,  35,  20,  15,   5,   1;
  1, 1, 11, 20, 45, 36,  84,  56,  70,  35,  21,  6,  1;
  1, 1, 13, 12, 66, 55, 165, 120, 210, 126, 126, 56, 28, 7, 1;
  ...
		

Crossrefs

Cf. A001906 (row sums).

Cf. A140737, A005408 (3rd column), A005843 (4th column), A000384 (5th column), A014105 (6th column), A000447 (7th column)

Programs

Maple

A140736 := proc(n,k)
    local X,r,c ;
    X := Matrix(2*n+1,2*n+1) ;
    for r from 1 to 2*n+1 do
    for c from 1 to 2*n+1 do
        if r = c then
            if type(r,'odd') then
                X[r,c] := 1 ;
            else
                X[r,c] := 0 ;
            end if ;
        elif r = c+1 or r=c+2 then
            X[r,c] := 1 ;
        end if;
    end do:
    end do:
    LinearAlgebra[MatrixPower](X,n) ;
    %[k,1] ;
end proc:
seq(seq( A140736(n,k),k=1..2*n+1),n=0..12) ; # R. J. Mathar, Nov 14 2023