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A123965
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Triangle read by rows: T(0,0)=1; T(n,k)=coefficient of x^k in the polynomial (-1)^n*p[n,x], where p[n,x] is the monic 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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5
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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, 17711, -76500, 143682
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| T(n,0)=fibonacci(2n+2)=A001906(n+1).
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 15 2010: (Start)
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 - 9x^2 + 25x - 21
relate to the octagon and are such that the product with k=1,2,3 =
(4.414213...,)*(3)*(1.585786...,) = 21. (End)
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REFERENCES
| Eric Weisstein's World of Mathematics, Tridiagonal Matrix, http://mathworld.wolfram.com/TridiagonalMatrix.html
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FORMULA
| a(n,m)=If[ n == m, 3, If[n == m - 1 || n == m + 1, -1, 0]]] p(n,x)=CharacteristicPolynomial(a(n,m)) p(n,x)->t(n,m)
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EXAMPLE
| Polynomials:
1
3 - x,
8 - 6 x + x^2,
21 - 25x + 9 x^2 - x^3,
55 - 90x + 51 x^2 - 12 x^3 + x^4,
144 - 300x + 234x^2 - 86x^3 + 15x^4 - x^5,
377 - 954 x + 951 x^2 - 480 x^3 + 130x^4 - 18 x^5 + x^6,
987 - 2939 x + 3573x^2 - 2305 x^3 + 855 x^4 - 183 x^5 + 21 x^6 - x^7
Triangular sequence:
{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
| 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
| Clear[M, T, d, a, x]; T[n_, m_] = If[ n == m, 3, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m], {n, 1, d}, {m, 1, d}]; Table[M[d], {d, 1, 10}]; Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{3}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a]
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CROSSREFS
| Cf. A123343.
Cf. A001906.
Cf. A152063 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 15 2010]
Sequence in context: A103247 A030523 * A124025 A125662 A005295 A077897
Adjacent sequences: A123962 A123963 A123964 * A123966 A123967 A123968
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KEYWORD
| sign,tabl
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AUTHOR
| Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 28 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 24 2006
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