A124576 Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (1,4,4,...) and super- and subdiagonals (1,1,1,...).

1, 1, 1, 2, 5, 1, 7, 23, 9, 1, 30, 108, 60, 13, 1, 138, 522, 361, 113, 17, 1, 660, 2587, 2079, 830, 182, 21, 1, 3247, 13087, 11733, 5581, 1579, 267, 25, 1, 16334, 67328, 65600, 35636, 12164, 2672, 368, 29, 1, 83662, 351246, 365364, 220308, 86964, 23220, 4173

Offset: 1

Author

Gary W. Adamson & Roger L. Bagula, Nov 05 2006

Keywords

• nonn
• tabl

Comments

Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+4*T(n-1,k)+T(n-1,k+1) for k>=1. - Philippe Deléham, Mar 27 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

Examples

			Row 3 is (2,5,1) because M[3]=[1,1,0;1,4,1;0,1,4] and M[3]^2=[2,5,1;5,18,8;1,8,17].
Triangle starts:
1;
1, 1;
2, 5, 1;
7, 23, 9, 1;
30, 108, 60, 13, 1;
138, 522, 361, 113, 17, 1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A124575, A124574, A052179, A227081 (row sums).

Programs

Maple

with(linalg): m:=proc(i,j) if i=1 and j=1 then 1 elif i=j then 4 elif abs(i-j)=1 then 1 else 0 fi end: for n from 3 to 11 do A[n]:=matrix(n,n,m): B[n]:=multiply(seq(A[n],i=1..n-1)) od: 1; 1,1; for n from 3 to 11 do seq(B[n][1,j],j=1..n) od; # yields sequence in triangular form
# alternative
A124576_row := proc(n)
    if n = 0 then
        return [1] ;
    else
        M := Matrix(n,n) ;
        M[1,1] := 1;
        for c from 2 to n do
            if c = 2 then
                M[1,c] := 1;
            else
                M[1,c] := 0;
            end if;
        end do:
        for r from 2 to n do
            for c from 1 to n do
                if r = c then
                    M[r,c] := 4;
                elif abs(r-c) = 1 then
                    M[r,c] := 1;
                else
                    M[r,c] := 0;
                end if;
            end do:
        end do:
        LinearAlgebra[MatrixPower](M,n-1) ;
        return [seq(%[1,r],r=1..n)] ;
    end if;
end proc:
for n from 0 to 10 do
    A124576_row(n) ;
    print(%) ;
end do: # R. J. Mathar, May 20 2025

Mathematica

M[n_] := SparseArray[{{1, 1} -> 1, Band[{2, 2}] -> 4, Band[{1, 2}] -> 1, Band[{2, 1}] -> 1}, {n, n}]; row[1] = {1}; row[n_] := MatrixPower[M[n], n-1] // First // Normal; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

Formula

Sum_{k=0..n} T(n,k)*(4*k+1) = 6^n. - Philippe Deléham, Mar 27 2007

Extensions

Edited by N. J. A. Sloane, Dec 04 2006