cp's OEIS Frontend

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).

%I A123965 #42 Feb 16 2025 08:33:03
%S A123965 1,3,-1,8,-6,1,21,-25,9,-1,55,-90,51,-12,1,144,-300,234,-86,15,-1,377,
%T A123965 -954,951,-480,130,-18,1,987,-2939,3573,-2305,855,-183,21,-1,2584,
%U A123965 -8850,12707,-10008,4740,-1386,245,-24,1,6765,-26195,43398,-40426,23373,-8715,2100,-316,27,-1
%N 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).
%C A123965 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
%H A123965 G. C. Greubel, <a href="/A123965/b123965.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A123965 J. Dombrowski, <a href="https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-114/issue-2/Tridiagonal-matrix-representations-of-cyclic-selfadjoint-operators/pjm/1102708710.full">Tridiagonal matrix representations of cyclic self-adjoint operators</a>, Pacif. J. Math. 114 (2): 324-334 (1984).
%H A123965 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TridiagonalMatrix.html">Tridiagonal Matrix</a>.
%F A123965 T(n, 0) = Fibonacci(2*n+2) = A001906(n+1).
%F A123965 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
%F A123965 From _G. C. Greubel_, Aug 20 2023: (Start)
%F A123965 T(n, k) = [x^k]( ChebyshevU(n, (3-x)/2) ).
%F A123965 Sum_{k=0..n} T(n, k) = n+1.
%F A123965 Sum_{k=0..n} (-1)^k*T(n, k) = A001353(n+1).
%F A123965 Sum_{k=0..floor(n/2)} T(n-k, k) = A000225(n+1).
%F A123965 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000244(n). (End)
%e A123965 Polynomials p(n, x):
%e A123965     1,
%e A123965     3 -     x,
%e A123965     8 -   6*x +     x^2,
%e A123965    21 -  25*x +   9*x^2 -     x^3,
%e A123965    55 -  90*x +  51*x^2 -  12*x^3 +    x^4,
%e A123965   144 - 300*x + 234*x^2 -  86*x^3 +  15*x^4 -    x^5,
%e A123965   377 - 954*x + 951*x^2 - 480*x^3 + 130*x^4 - 18*x^5 + x^6,
%e A123965   ...
%e A123965 Triangle begins:
%e A123965      1;
%e A123965      3,     -1;
%e A123965      8,     -6,     1;
%e A123965     21,    -25,     9,     -1;
%e A123965     55,    -90,    51,    -12,     1;
%e A123965    144,   -300,   234,    -86,    15,    -1;
%e A123965    377,   -954,   951,   -480,   130,   -18,    1;
%e A123965    987,  -2939,  3573,  -2305,   855,  -183,   21,   -1;
%e A123965   2584,  -8850, 12707, -10008,  4740, -1386,  245,  -24,  1;
%e A123965   6765, -26195, 43398, -40426, 23373, -8715, 2100, -316, 27, -1;
%e A123965   ...
%p A123965 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
%t A123965 (* First program *)
%t A123965 T[n_, m_]:= If[n==m, 3, If[n==m-1 || n==m+1, -1, 0]];
%t A123965 M[d_]:= Table[T[n, m], {n,d}, {m,d}];
%t A123965 Table[M[d], {d,10}];
%t A123965 Table[Det[M[d] - x*IdentityMatrix[d]], {d,10}];
%t A123965 Join[{{3}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d,10}]]//Flatten
%t A123965 (* Second program *)
%t A123965 Table[CoefficientList[ChebyshevU[n, (3-x)/2], x], {n,0,12}]//Flatten (* _G. C. Greubel_, Aug 20 2023 *)
%o A123965 (Magma)
%o A123965 m:=12;
%o A123965 p:= func< n,x | Evaluate(ChebyshevU(n+1), (3-x)/2) >;
%o A123965 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A123965 A123965:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
%o A123965 [A123965(n,k): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Aug 20 2023
%o A123965 (SageMath)
%o A123965 def A123965(n,k): return ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k]
%o A123965 flatten([[A123965(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Aug 20 2023
%Y A123965 Cf. A000045, A000225, A000244, A001353, A001906, A123343, A125662 (absolute values), A152063.
%K A123965 sign,tabl
%O A123965 0,2
%A A123965 _Gary W. Adamson_ and _Roger L. Bagula_, Oct 28 2006
%E A123965 Edited by _N. J. A. Sloane_, Nov 24 2006