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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.

%I A126570 #24 Jun 30 2025 09:57:42
%S A126570 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,
%T A126570 6,1,14,78,220,330,252,84,9,1,16,105,364,715,792,462,120,8,1,18,136,
%U A126570 560,1365,2002,1716,792,165,11,1,20,171,816,2380,4368,5005,3432,1287,220,10
%N 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.
%D A126570 William G. Harter, Physics Department, University of Arkansas; personal communication.
%F A126570 It seems that for n > 2, T(n, k) = binomial(2n+1-k, k) - [k=n] * (-1)^n. - _Andrei Zabolotskii_, Jun 29 2025
%e A126570 First few rows of the triangle are:
%e A126570   1;
%e A126570   1, 2;
%e A126570   1, 4, 3;
%e A126570   1, 6, 10, 5;
%e A126570   1, 8, 21, 20, 4;
%e A126570   1, 10, 36, 56, 35, 7;
%e A126570   1, 12, 55, 120, 126, 56, 6;
%e A126570   ...
%e A126570 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.
%t A126570 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 *)
%Y A126570 Cf. A152060, A053123.
%K A126570 nonn,tabl
%O A126570 0,3
%A A126570 _Gary W. Adamson_, Dec 28 2006
%E A126570 Edited by _N. J. A. Sloane_, Aug 10 2019
%E A126570 Edited and extended by _Andrei Zabolotskii_, Jun 29 2025