A123965 Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^k in the polynomial (-1)^n*p(n,x), where p(n,x) is the characteristic polynomial of the n X n tridiagonal matrix with 3's on the main diagonal and -1's on the super- and subdiagonal (n >= 1; 0 <= k <= n).

1, 3, -1, 8, -6, 1, 21, -25, 9, -1, 55, -90, 51, -12, 1, 144, -300, 234, -86, 15, -1, 377, -954, 951, -480, 130, -18, 1, 987, -2939, 3573, -2305, 855, -183, 21, -1, 2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1, 6765, -26195, 43398, -40426, 23373, -8715, 2100, -316, 27, -1

Offset: 0

Author

Gary W. Adamson and Roger L. Bagula, Oct 28 2006

Keywords

• sign
• tabl

Comments

Reversed polynomials = bisection of A152063: (1; 1,3; 1,6,8; 1,9,25,21; ...) having the following property: even-indexed Fibonacci numbers = Product_{k=1..n-2/2} (1 + 4*cos^2 k*Pi/n); n relating to regular polygons with an even number of edges. Example: The roots to x^3 - 9*x^2 + 25*x - 21 relate to the octagon and are such that the product with k=1,2,3 = (4.414213...)*(3)*(1.585786...) = 21. - Gary W. Adamson, Aug 15 2010

Examples

			Polynomials p(n, x):
    1,
    3 -     x,
    8 -   6*x +     x^2,
   21 -  25*x +   9*x^2 -     x^3,
   55 -  90*x +  51*x^2 -  12*x^3 +    x^4,
  144 - 300*x + 234*x^2 -  86*x^3 +  15*x^4 -    x^5,
  377 - 954*x + 951*x^2 - 480*x^3 + 130*x^4 - 18*x^5 + x^6,
  ...
Triangle begins:
     1;
     3,     -1;
     8,     -6,     1;
    21,    -25,     9,     -1;
    55,    -90,    51,    -12,     1;
   144,   -300,   234,    -86,    15,    -1;
   377,   -954,   951,   -480,   130,   -18,    1;
   987,  -2939,  3573,  -2305,   855,  -183,   21,   -1;
  2584,  -8850, 12707, -10008,  4740, -1386,  245,  -24,  1;
  6765, -26195, 43398, -40426, 23373, -8715, 2100, -316, 27, -1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• J. Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, Pacif. J. Math. 114 (2): 324-334 (1984).
• Eric Weisstein's World of Mathematics, Tridiagonal Matrix.

Crossrefs

Cf. A000045, A000225, A000244, A001353, A001906, A123343, A125662 (absolute values), A152063.

Programs

Magma

m:=12;
p:= func< n,x | Evaluate(ChebyshevU(n+1), (3-x)/2) >;
R:=PowerSeriesRing(Integers(), m+2);
A123965:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
[A123965(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023

Maple

with(linalg): a:=proc(i,j) if j=i then 3 elif abs(i-j)=1 then -1 else 0 fi end: for n from 1 to 10 do p[n]:=(-1)^n*charpoly(matrix(n,n,a),x) od: 1; for n from 1 to 10 do seq(coeff(p[n],x,j),j=0..n) od; # yields sequence in triangular form

Mathematica

(* First program *)
T[n_, m_]:= If[n==m, 3, If[n==m-1 || n==m+1, -1, 0]];
M[d_]:= Table[T[n, m], {n,d}, {m,d}];
Table[M[d], {d,10}];
Table[Det[M[d] - x*IdentityMatrix[d]], {d,10}];
Join[{{3}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d,10}]]//Flatten
(* Second program *)
Table[CoefficientList[ChebyshevU[n, (3-x)/2], x], {n,0,12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)

SageMath

def A123965(n,k): return ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k]
flatten([[A123965(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023

Formula

T(n, 0) = Fibonacci(2*n+2) = A001906(n+1).

Equals coefficients of the polynomials p(n,x) = (3-x)*p(n-1,x) - p(n-2,x), with p(0, x) = 1, p(1, x) = 3-x. - Roger L. Bagula, Oct 31 2006

From G. C. Greubel, Aug 20 2023: (Start)

T(n, k) = [x^k]( ChebyshevU(n, (3-x)/2) ).

Sum_{k=0..n} T(n, k) = n+1.

Sum_{k=0..n} (-1)^k*T(n, k) = A001353(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A000225(n+1).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000244(n). (End)

Extensions

Edited by N. J. A. Sloane, Nov 24 2006