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A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.

%I A124029 #25 Aug 21 2023 08:24:53
%S A124029 1,4,-1,15,-8,1,56,-46,12,-1,209,-232,93,-16,1,780,-1091,592,-156,20,
%T A124029 -1,2911,-4912,3366,-1200,235,-24,1,10864,-21468,17784,-8010,2120,
%U A124029 -330,28,-1,40545,-91824,89238,-48624,16255,-3416,441,-32,1,151316,-386373,430992,-275724,111524,-29589,5152,-568,36,-1
%N A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.
%C A124029 The matrices are {4} if n=1, {{4,-1},{-1,4}} if n=2, {{4,-1,0},{-1,4,-1},{0,-1,4}} if n=3 etc. The empty matrix at n=0 has an empty product (determinant) with assigned value =1.
%C A124029 Riordan array (1/(1-4*x+x^2), -x/(1-4*x+x^2)). - _Philippe Deléham_, Mar 04 2016
%H A124029 G. C. Greubel, <a href="/A124029/b124029.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A124029 Joanne Dombrowski, <a href="http://projecteuclid.org/euclid.pjm/1102703882">Tridiagonal matrix representations of cyclic self-adjoint operators</a>, Pacific J. Math. 114, no. 2 (1984), 325-334.
%F A124029 T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (4-x)*p(n-1, x) - p(n-2, x), p(0, x) = 1, p(1, x) = 4-x.
%F A124029 From _G. C. Greubel_, Aug 20 2023: (Start)
%F A124029 T(n, k) = [x^k]( ChebyshevU(n, (4-x)/2) ).
%F A124029 Sum_{k=0..n} T(n, k) = A001906(n+1).
%F A124029 Sum_{k=0..n} (-1)^k*T(n, k) = A004254(n+1).
%F A124029 Sum_{k=0..floor(n/2)} T(n-k, k) = A007070(n).
%F A124029 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000302(n).
%F A124029 T(n, n) = (-1)^n.
%F A124029 T(n, n-1) = 4*A181983(n), n >= 1.
%F A124029 T(n, n-2) = (-1)^n*A139278(n-1), n >= 2.
%F A124029 T(n, 0) = A001353(n+1). (End)
%e A124029 Triangle begins as:
%e A124029       1;
%e A124029       4,     -1;
%e A124029      15,     -8,     1;
%e A124029      56,    -46,    12,    -1;
%e A124029     209,   -232,    93,   -16,    1;
%e A124029     780,  -1091,   592,  -156,   20,   -1;
%e A124029    2911,  -4912,  3366, -1200,  235,  -24,  1;
%e A124029   10864, -21468, 17784, -8010, 2120, -330, 28, -1;
%p A124029 A123966x := proc(n,x)
%p A124029     local A,r,c ;
%p A124029     A := Matrix(1..n,1..n) ;
%p A124029     for r from 1 to n do
%p A124029     for c from 1 to n do
%p A124029             A[r,c] :=0 ;
%p A124029         if r = c then
%p A124029             A[r,c] := A[r,c]+4 ;
%p A124029         elif abs(r-c)= 1 then
%p A124029             A[r,c] :=  A[r,c]-1 ;
%p A124029         end if;
%p A124029     end do:
%p A124029     end do:
%p A124029     (-1)^n*LinearAlgebra[CharacteristicPolynomial](A,x) ;
%p A124029 end proc;
%p A124029 A123966 := proc(n,k)
%p A124029     coeftayl( A123966x(n,x),x=0,k) ;
%p A124029 end proc:
%p A124029 seq(seq(A123966(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Dec 06 2011
%t A124029 (* Matrix version*)
%t A124029 k = 4;
%t A124029 T[n_, m_, d_]:= If[n==m, k, If[n==m-1 || n==m+1, -1, 0]];
%t A124029 M[d_]:= Table[T[n, m, d], {n,d}, {m,d}];
%t A124029 Table[M[d], {d,10}]
%t A124029 Table[Det[M[d]], {d,10}]
%t A124029 Table[Det[M[d] - x*IdentityMatrix[d]], {d,10}]
%t A124029 Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d,10}]]//Flatten
%t A124029 (* Recursive Polynomial form*)
%t A124029 p[0, x]= 1; p[1, x]= (4-x); p[k_, x_]:= p[k, x]= (4-x)*p[k-1, x] - p[k -2, x];
%t A124029 Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
%t A124029 (* Additional program *)
%t A124029 Table[CoefficientList[ChebyshevU[n, (4-x)/2], x], {n,0,12}]//Flatten (* _G. C. Greubel_, Aug 20 2023 *)
%o A124029 (Magma)
%o A124029 m:=12;
%o A124029 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A124029 A124029:= func< n,k | Coefficient(R!( Evaluate(ChebyshevU(n+1), (4-x)/2) ), k) >;
%o A124029 [A124029(n,k): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Aug 20 2023
%o A124029 (SageMath)
%o A124029 def A124029(n,k): return ( chebyshev_U(n, (4-x)/2) ).series(x, n+2).list()[k]
%o A124029 flatten([[A124029(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Aug 20 2023
%Y A124029 Cf. A000302, A001353, A001906, A004254, A007070, A123966.
%Y A124029 Cf. A139278, A159764, A181983, A207823 (absolute values).
%K A124029 tabl,sign
%O A124029 0,2
%A A124029 _Gary W. Adamson_ and _Roger L. Bagula_, Nov 01 2006