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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.
4
1, 4, -1, 15, -8, 1, 56, -46, 12, -1, 209, -232, 93, -16, 1, 780, -1091, 592, -156, 20, -1, 2911, -4912, 3366, -1200, 235, -24, 1, 10864, -21468, 17784, -8010, 2120, -330, 28, -1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36, -1
OFFSET
0,2
COMMENTS
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.
Riordan array (1/(1-4*x+x^2), -x/(1-4*x+x^2)). - Philippe Deléham, Mar 04 2016
LINKS
Joanne Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334.
FORMULA
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.
From G. C. Greubel, Aug 20 2023: (Start)
T(n, k) = [x^k]( ChebyshevU(n, (4-x)/2) ).
Sum_{k=0..n} T(n, k) = A001906(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A004254(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007070(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000302(n).
T(n, n) = (-1)^n.
T(n, n-1) = 4*A181983(n), n >= 1.
T(n, n-2) = (-1)^n*A139278(n-1), n >= 2.
T(n, 0) = A001353(n+1). (End)
EXAMPLE
Triangle begins as:
1;
4, -1;
15, -8, 1;
56, -46, 12, -1;
209, -232, 93, -16, 1;
780, -1091, 592, -156, 20, -1;
2911, -4912, 3366, -1200, 235, -24, 1;
10864, -21468, 17784, -8010, 2120, -330, 28, -1;
MAPLE
A123966x := proc(n, x)
local A, r, c ;
A := Matrix(1..n, 1..n) ;
for r from 1 to n do
for c from 1 to n do
A[r, c] :=0 ;
if r = c then
A[r, c] := A[r, c]+4 ;
elif abs(r-c)= 1 then
A[r, c] := A[r, c]-1 ;
end if;
end do:
end do:
(-1)^n*LinearAlgebra[CharacteristicPolynomial](A, x) ;
end proc;
A123966 := proc(n, k)
coeftayl( A123966x(n, x), x=0, k) ;
end proc:
seq(seq(A123966(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Dec 06 2011
MATHEMATICA
(* Matrix version*)
k = 4;
T[n_, m_, d_]:= If[n==m, k, If[n==m-1 || n==m+1, -1, 0]];
M[d_]:= Table[T[n, m, d], {n, d}, {m, d}];
Table[M[d], {d, 10}]
Table[Det[M[d]], {d, 10}]
Table[Det[M[d] - x*IdentityMatrix[d]], {d, 10}]
Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 10}]]//Flatten
(* Recursive Polynomial form*)
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];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
(* Additional program *)
Table[CoefficientList[ChebyshevU[n, (4-x)/2], x], {n, 0, 12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)
PROG
(Magma)
m:=12;
R<x>:=PowerSeriesRing(Integers(), m+2);
A124029:= func< n, k | Coefficient(R!( Evaluate(ChebyshevU(n+1), (4-x)/2) ), k) >;
[A124029(n, k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023
(SageMath)
def A124029(n, k): return ( chebyshev_U(n, (4-x)/2) ).series(x, n+2).list()[k]
flatten([[A124029(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023
CROSSREFS
Cf. A139278, A159764, A181983, A207823 (absolute values).
Sequence in context: A080419 A095307 A159764 * A207823 A056920 A123382
KEYWORD
tabl,sign
AUTHOR
STATUS
approved