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%I #10 Sep 21 2015 03:52:41
%S 1,2,-1,-3,2,1,-4,6,2,-1,5,-10,-9,2,1,6,-19,-16,12,2,-1,-7,28,42,-22,
%T -15,2,1,-8,44,68,-74,-28,18,2,-1,9,-60,-138,126,115,-34,-21,2,1,10,
%U -85,-208,316,202,-165,-40,24,2,-1,-11,110,363,-506,-605,296,224,-46,-27,2,1
%N Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n matrix: 2 on the main antidiagonal, -1 on the adjacent sub-antidiagonals and 0 otherwise.
%C We start from tri-antidiagonal variants of the Cartan A-n group matrix. For n=1 this is {2}, for n=2 this is {{-1,2},{2,-1}}, for n=3 {{0,-1,2},{-1,2,-1},{2,-1,0}}, for n =4 {{0,0,-1,2},{0,-1,2,-1},{-1,2,-1,0},{2,-1,0,0}} etc. The n-th row of the triangle are the expansion coefficients of the characteristic polynomial.
%C For n=0, the empty product of the empty matrix is assigned the value T(0,0)=1.
%C Row sums (characteristic polynomials evaluated at x=0) are 1, 1, 0, 3, -11, -16, 29, 21, 0, 55, -199, -288, 521, 377, 0, 987, -3571, -5168, 9349, 6765, 0, ... (see A038150).
%e 1;
%e 2, -1;
%e -3,2, 1;
%e -4, 6, 2, -1;
%e 5, -10, -9, 2, 1;
%e 6, -19, -16, 12, 2, -1;
%e -7,28, 42, -22, -15, 2, 1;
%e -8, 44, 68, -74, -28,18, 2, -1;
%e 9, -60, -138, 126, 115, -34, -21, 2, 1;
%e 10, -85, -208,316, 202, -165, -40, 24, 2, -1;
%e -11, 110, 363, -506, -605, 296, 224, -46, -27, 2, 1;
%p A136451x := proc(n,x)
%p local A,r,c ;
%p A := Matrix(1..n,1..n) ;
%p for r from 1 to n do
%p for c from 1 to n do
%p A[r,c] :=0 ;
%p if r+c = 1+n then
%p A[r,c] := A[r,c]+2 ;
%p elif abs(r+c-1-n)= 1 then
%p A[r,c] := A[r,c]-1 ;
%p end if;
%p end do:
%p end do:
%p (-1)^n*LinearAlgebra[CharacteristicPolynomial](A,x) ;
%p end proc;
%p A136451 := proc(n,k)
%p coeftayl( A136451x(n,x),x=0,k) ;
%p end proc:
%p seq(seq(A136451(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Dec 04 2011
%t H[n_] := Table[Table[If[i + j - 1 == n, 2,If[i + j - 1 == n + 1, -1, If[i + j - 1 == n - 1, -1, 0]]], {i, 1, n}], {j, 1, n}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[H[n], x], x], {n, 1, 10}]]; Flatten[a']
%Y Cf. A124018 (variant), A005993 (column k=1), A061927 (bisection column k=2).
%K tabl,sign
%O 0,2
%A _Roger L. Bagula_, Mar 19 2008
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