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A123971
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Tridiagonal matrices of the type 3 center with terminal 2 as characteristic polynomial triangular sequence.
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0
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2, 2, -1, 5, -5, 1, 13, -19, 8, -1, 34, -65, 42, -11, 1, 89, -210, 183, -74, 14, -1, 233, -654, 717, -394, 115, -17, 1, 610, -1985, 2622, -1825, 725, -165, 20, -1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1, 4181, -17345, 30691, -30418, 18633, -7329, 1855, -292, 26, -1, 10946, -50305, 100284, -114242
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Major roots are odd Tutte-Beraha constants plus one: 2., 3.61803, 4.24698, 4.53209, 4.68251, 4.77091, 4.82709, 4.86494, 4.89163, 4.91115} Matrices: {{2}}, {{3, -1}, {-1, 2}}, {{3, -1, 0}, {-1, 3, -1}, {0, -1, 2}}, {{3, -1, 0, 0}, {-1, 3, -1, 0}, {0, -1, 3, -1}, {0, 0, -1, 2}}, {{3, -1, 0, 0, 0}, {-1, 3, -1, 0, 0}, {0, -1, 3, -1, 0}, {0, 0, -1, 3, -1}, {0, 0, 0, -1, 2}}, {{3, -1, 0, 0, 0, 0}, {-1, 3, -1, 0, 0, 0}, {0, -1, 3, -1, 0, 0}, {0, 0, -1, 3, -1, 0}, {0, 0, 0, -1, 3, -1}, {0, 0, 0, 0, -1, 2}} The determinants are increasing: {2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946}
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FORMULA
| M(n,m,d)=If[ n == m && n < d && m < d, k, If[n == m - 1 || n == m + 1, -1, If[n == m == d, k - 1, 0]]];
T(0,0)=2 (=1 would be natural; see Sloane's comment below); T(n,k) = (-1)^k*sum(m=k,n, binomial(m,k)*binomial(m+n,2*m) ). - Wadim Zudilin, Jan 11 2012
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EXAMPLE
| Triangular sequence:
{2},
{2, -1},
{5, -5, 1},
{13, -19, 8, -1},
{34, -65, 42, -11,1},
{89, -210, 183, -74, 14, -1}
{233, -654, 717, -394, 115, -17, 1}
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MATHEMATICA
| Mathematica ( general k th center) Clear[M, T, d, a, x, k] k = 3 T[n_, m_, d_] := If[ n == m && n < d && m < d, k, If[n == m - 1 || n == m + 1, -1, If[n == m == d, k - 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] Table[NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x], {d, 1, 10}] Table[x /. NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x][[d]], {d, 1, 10}]
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CROSSREFS
| Unsigned version equals unsigned A126124
Sequence in context: A158068 A176265 A187307 * A204851 A114292 A178518
Adjacent sequences: A123968 A123969 A123970 * A123972 A123973 A123974
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 30 2006
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EXTENSIONS
| Looking at the triangle suggests that the very first term should be 1, not 2. - N. J. A. Sloane (njas(AT)research.att.com), Nov 01, 2006
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