OFFSET
1,5
COMMENTS
Matrices {{1}}, {{1, -1}, {-2, 2}}, {{1, -1, 0}, {-2, 2, -1}, {1, -2, 2}}, {{1, -1, 0, 0}, {-2, 2, -1, 0}, {1, -2, 2, -1}, {0, 1, -2, 2}}, {{1, -1, 0, 0, 0}, {-2, 2, -1, 0, 0}, {1, -2, 2, -1, 0}, {0, 1, -2, 2, -1}, {0, 0, 1, -2, 2}} Large root sequence: a0 = Table[x /. NSolve[CharacteristicPolynomial[Xn[d], x] == 0, x][[d]], {d, 1, 20}] {1., 3., 3.91223, 4.37167, 4.62826, 4.78478, 4.88683, 4.95691, 5.00703, 5.0441, 5.07226, 5.09415, 5.1115, 5.12547, 5.1369, 5.14636, 5.15428, 5.16097, 5.16668, 5.17159} Determinant sequence is: Table[Det[Xn[d]], {d, 1, 20}] {1,0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0}
FORMULA
Steinbach(n,m)=If[n + m - 1 > d, 0, 1]; tri-Antidiagonal_1(n,m)=If[n + m - 1 == d, 1, If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]]; X(n,m)=Steinbach(n,m)^(-1).tri-Antidiagonal2(n,n)
EXAMPLE
triangular sequence:
{1},
{1, -1},
{0, -3, 1},
{-1, -4,5, -1},
{-1, -2, 12, -7, 1},
{0, 2, 17, -24, 9, -1},
{1, 4,13, -52, 40, -11,1},
{1, 1, 0, -76, 115, -60, 13, -1},
{0, -5, -10, -72, 235, -214, 84, -15,1}
MATHEMATICA
An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Bn[d_] := Table[If[n + m - 1 == d, 1, If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]], {n, 1, d}, {m, 1, d}]; Xn[d_] := MatrixPower[An[d], -1].Bn[d]; a = Join[Xn[1], Table[CoefficientList[CharacteristicPolynomial[Xn[d], x], x], {d, 1, 20}]]; Flatten[%]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula and Gary W. Adamson, Oct 31 2006
STATUS
approved