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A124029
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Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.
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2
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1, 4, -1, 15, -8, 1, 56, -46, 12, -1, 209, -232, 93, -16, 1, 780, -1091, 592, -156, 20, -1, 2911, -4912, 3366, -1200, 235, -24, 1, 10864, -21468, 17784, -8010, 2120, -330, 28, -1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The matrices are {4} if n=1, {{4,-1},{-1,4}} if n=2, {{4,-1,0},{-1,4,-1},{0,-1,4}} if n=3 etc. The empty matrix at n=0 has an empty product (determinant) with assigned value =1.
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REFERENCES
| Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334
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EXAMPLE
| 1;
4, -1;
15, -8, 1;
56, -46,12, -1;
209, -232, 93, -16, 1;
780, -1091, 592, -156, 20, -1;
2911, -4912, 3366, -1200, 235, -24, 1;
10864, -21468, 17784, -8010, 2120, -330, 28, -1;
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MAPLE
| A123966x := proc(n, x)
local A, r, c ;
A := Matrix(1..n, 1..n) ;
for r from 1 to n do
for c from 1 to n do
A[r, c] :=0 ;
if r = c then
A[r, c] := A[r, c]+4 ;
elif abs(r-c)= 1 then
A[r, c] := A[r, c]-1 ;
end if;
end do:
end do:
(-1)^n*LinearAlgebra[CharacteristicPolynomial](A, x) ;
end proc;
A123966 := proc(n, k)
coeftayl( A123966x(n, x), x=0, k) ;
end proc:
seq(seq(A123966(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Dec 06 2011
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MATHEMATICA
| (* Matrix version*) k = 4; T[n_, m_, d_] := If[ n == m, k, If[n == m - 1 || n == m + 1, -1, 0]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a] (* Recursive Polynomial form*) b[k_] = 4; a[k_] = -1; p[0, x] = 1; p[1, x] = (x - b[1])/a[1]; p[k_, x_] := p[k, x] = ((x - b[k - 1])*p[k - 1, x] - a[k - 2] *p[k - 2, x])/a[k - 1; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]
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CROSSREFS
| Cf. A123966, A159764.
Sequence in context: A080419 A095307 A159764 * A056920 A123382 A197653
Adjacent sequences: A124026 A124027 A124028 * A124030 A124031 A124032
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KEYWORD
| tabl,sign
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AUTHOR
| Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Nov 01 2006
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