A204842 Triangle by rows relating to A081696.

1, 1, 2, 3, 4, 2, 9, 12, 6, 2, 29, 38, 20, 8, 2, 97, 126, 68, 30, 10, 2, 333, 430, 236, 110, 42, 12, 2, 1165, 1498, 832, 402, 166, 56, 14, 2, 4135, 5300, 2970, 1472, 640, 238, 72, 16, 2, 14845, 18980, 10710, 5410, 2440, 968, 328, 90, 18, 2

Offset: 0

Author

Gary W. Adamson, Jan 19 2012

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle =
1;
1, 2;
3, 4, 2;
9, 12, 6, 2;
29, 38, 20, 8, 2;
97, 126, 68, 30, 10, 2;
333, 430, 236, 110, 42, 12, 2;
1165, 1498, 832, 402, 166, 56, 14, 2;
4135, 5300, 2970, 1472, 640, 238, 72, 16, 2;
14845, 18980, 10710, 5410, 2440, 968, 328, 90, 18, 2;
...
Top row of M^3 = [9, 12, 6, 2, 0, 0, 0,...]
		

Crossrefs

Cf. A081696 (first column and also row sums).

Programs

Maple

A204842T := proc(n,k)
    if n =0 and k =1 then
        2;
    elif k <0 or k >n+1 then
        0;
    else
        1;
    end if ;
end proc:
A204842 := proc(n,k)
    local A;
    A := Matrix(n+1,n+1) ;
    for row from 1 to n+1 do
    for col from 1 to n+1 do
            A[row, col] := A204842T(row-1,col-1) ;
    end do:
    end do:
    Mn := LinearAlgebra[MatrixPower](A , n);
    Mn[1,k+1] ;
end proc:
for n from 0 to 10 do
    for k from 0 to n do
    printf("%d ",A204842(n,k)) ;
    end do:
    printf("\n") ;
end do:

Formula

n-th row of the triangle is the top row of M^n, where M is the following infinite square production matrix:

1, 2, 0, 0, 0, 0,...

1, 1, 1, 0, 0, 0,...

1, 1, 1, 1, 0, 0,...

1, 1, 1, 1, 1, 0,...

1, 1, 1, 1, 1, 1,...

...

Conjecture: T(n,1) = 2*A109262(n). T(n,2)=2*A109263(n). - R. J. Mathar, Jul 21 2015