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

1,4

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

Links

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

Formula

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

Example

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;

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 *)

Crossrefs

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

Sequence in context: A258067 A240241 A019510 * A283556 A340198 A362151

Adjacent sequences: A124573 A124574 A124575 * A124577 A124578 A124579

Keyword

nonn,tabl

Author

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

Extensions

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

Status

approved