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

0,7

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

Links

Table of n, a(n) for n=0..63.

Example

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;
  ...

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

Crossrefs

Cf. A001906 (row sums).

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

Sequence in context: A106689 A348177 A027082 * A284993 A292068 A378066

Adjacent sequences: A140733 A140734 A140735 * A140737 A140738 A140739

Keyword

nonn,tabf,easy

Author

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

Status

approved