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%I #41 Apr 04 2024 10:11:13
%S 1,3,-1,8,-6,1,21,-25,9,-1,55,-90,51,-12,1,144,-300,234,-86,15,-1,377,
%T -954,951,-480,130,-18,1,987,-2939,3573,-2305,855,-183,21,-1,2584,
%U -8850,12707,-10008,4740,-1386,245,-24,1,6765,-26195,43398,-40426,23373,-8715,2100,-316,27,-1
%N 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 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 G. C. Greubel, <a href="/A123965/b123965.txt">Rows n = 0..50 of the triangle, flattened</a>
%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TridiagonalMatrix.html">Tridiagonal Matrix</a>.
%F T(n, 0) = Fibonacci(2*n+2) = A001906(n+1).
%F 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 From _G. C. Greubel_, Aug 20 2023: (Start)
%F T(n, k) = [x^k]( ChebyshevU(n, (3-x)/2) ).
%F Sum_{k=0..n} T(n, k) = n+1.
%F Sum_{k=0..n} (-1)^k*T(n, k) = A001353(n+1).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A000225(n+1).
%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000244(n). (End)
%e Polynomials p(n, x):
%e 1,
%e 3 - x,
%e 8 - 6*x + x^2,
%e 21 - 25*x + 9*x^2 - x^3,
%e 55 - 90*x + 51*x^2 - 12*x^3 + x^4,
%e 144 - 300*x + 234*x^2 - 86*x^3 + 15*x^4 - x^5,
%e 377 - 954*x + 951*x^2 - 480*x^3 + 130*x^4 - 18*x^5 + x^6,
%e ...
%e Triangle begins:
%e 1;
%e 3, -1;
%e 8, -6, 1;
%e 21, -25, 9, -1;
%e 55, -90, 51, -12, 1;
%e 144, -300, 234, -86, 15, -1;
%e 377, -954, 951, -480, 130, -18, 1;
%e 987, -2939, 3573, -2305, 855, -183, 21, -1;
%e 2584, -8850, 12707, -10008, 4740, -1386, 245, -24, 1;
%e 6765, -26195, 43398, -40426, 23373, -8715, 2100, -316, 27, -1;
%e ...
%p 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 (* First program *)
%t T[n_, m_]:= If[n==m, 3, If[n==m-1 || n==m+1, -1, 0]];
%t M[d_]:= Table[T[n, m], {n,d}, {m,d}];
%t Table[M[d], {d,10}];
%t Table[Det[M[d] - x*IdentityMatrix[d]], {d,10}];
%t Join[{{3}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d,10}]]//Flatten
%t (* Second program *)
%t Table[CoefficientList[ChebyshevU[n, (3-x)/2], x], {n,0,12}]//Flatten (* _G. C. Greubel_, Aug 20 2023 *)
%o (Magma)
%o m:=12;
%o p:= func< n,x | Evaluate(ChebyshevU(n+1), (3-x)/2) >;
%o R<x>:=PowerSeriesRing(Integers(), m+2);
%o A123965:= func< n,k | Coefficient(R!( p(n,x) ), k) >;
%o [A123965(n,k): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Aug 20 2023
%o (SageMath)
%o def A123965(n,k): return ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k]
%o flatten([[A123965(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Aug 20 2023
%Y Cf. A000045, A000225, A000244, A001353, A001906, A123343, A125662 (absolute values), A152063.
%K sign,tabl
%O 0,2
%A _Gary W. Adamson_ and _Roger L. Bagula_, Oct 28 2006
%E Edited by _N. J. A. Sloane_, Nov 24 2006
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