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A123965
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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).
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6
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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)
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EXAMPLE
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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;
...
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MAPLE
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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
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MATHEMATICA
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(* 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 *)
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PROG
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(Magma)
m:=12;
p:= func< n, x | Evaluate(ChebyshevU(n+1), (3-x)/2) >;
R<x>:=PowerSeriesRing(Integers(), m+2);
A123965:= func< n, k | Coefficient(R!( p(n, x) ), k) >;
(SageMath)
def A123965(n, k): return ( chebyshev_U(n, (3-x)/2) ).series(x, n+2).list()[k]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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