A126570 Triangle read by rows: row n gives coefficients (ignoring the alternating signs) of the characteristic polynomial of the n X n matrix with 2's in the main diagonal, 1's in the super- and subdiagonals, and 1 in the upper-right corner, with other elements zeros.

1, 1, 2, 1, 4, 3, 1, 6, 10, 5, 1, 8, 21, 20, 4, 1, 10, 36, 56, 35, 7, 1, 12, 55, 120, 126, 56, 6, 1, 14, 78, 220, 330, 252, 84, 9, 1, 16, 105, 364, 715, 792, 462, 120, 8, 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 11, 1, 20, 171, 816, 2380, 4368, 5005, 3432, 1287, 220, 10

Offset: 0

Author

Gary W. Adamson, Dec 28 2006

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle are:
  1;
  1, 2;
  1, 4, 3;
  1, 6, 10, 5;
  1, 8, 21, 20, 4;
  1, 10, 36, 56, 35, 7;
  1, 12, 55, 120, 126, 56, 6;
  ...
Charpoly of the 4 X 4 matrix [2,1,0,1; 1,2,1,0; 0,1,2,1; 0,0,1,2] = x^4 - 8*x^3 + 21*x^2 - 20*x + 4; with a root sqrt(3)+2.
		

References

• William G. Harter, Physics Department, University of Arkansas; personal communication.

Crossrefs

Cf. A152060, A053123.

Programs

Mathematica

M[i_,j_,n_]:=If[i==j,2,If[Abs[i-j]==1,1,If[j==n&&i==1,1,0]]]; row[0]=1; row[n_]:=Reverse[Abs[CoefficientList[CharacteristicPolynomial[Table[M[i,j,n],{i,n},{j,n}],x],x]]]; Array[row,11,0]//Flatten (* Stefano Spezia, Jun 30 2025 *)

Formula

It seems that for n > 2, T(n, k) = binomial(2n+1-k, k) - [k=n] * (-1)^n. - Andrei Zabolotskii, Jun 29 2025

Extensions

Edited by N. J. A. Sloane, Aug 10 2019

Edited and extended by Andrei Zabolotskii, Jun 29 2025